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Metamath Proof Explorer

Theorem oeordsuc

Description: Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of TakeutiZaring p. 68. (Contributed by NM, 7-Jan-2005)

Ref Expression
Assertion oeordsuc ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 onelon ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ 𝐵 ) → 𝐴 ∈ On )
2 1 ex ⊢ ( 𝐵 ∈ On → ( 𝐴 ∈ 𝐵 → 𝐴 ∈ On ) )
3 2 adantr ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → 𝐴 ∈ On ) )
4 oewordri ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( 𝐴 ↑o 𝐶 ) ⊆ ( 𝐵 ↑o 𝐶 ) ) )
5 4 3adant1 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( 𝐴 ↑o 𝐶 ) ⊆ ( 𝐵 ↑o 𝐶 ) ) )
6 oecl ⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ↑o 𝐶 ) ∈ On )
7 6 3adant2 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ↑o 𝐶 ) ∈ On )
8 oecl ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 ↑o 𝐶 ) ∈ On )
9 8 3adant1 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 ↑o 𝐶 ) ∈ On )
10 simp1 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → 𝐴 ∈ On )
11 omwordri ⊢ ( ( ( 𝐴 ↑o 𝐶 ) ∈ On ∧ ( 𝐵 ↑o 𝐶 ) ∈ On ∧ 𝐴 ∈ On ) → ( ( 𝐴 ↑o 𝐶 ) ⊆ ( 𝐵 ↑o 𝐶 ) → ( ( 𝐴 ↑o 𝐶 ) ·o 𝐴 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ) )
12 7 9 10 11 syl3anc ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ↑o 𝐶 ) ⊆ ( 𝐵 ↑o 𝐶 ) → ( ( 𝐴 ↑o 𝐶 ) ·o 𝐴 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ) )
13 5 12 syld ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( ( 𝐴 ↑o 𝐶 ) ·o 𝐴 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ) )
14 oesuc ⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ↑o suc 𝐶 ) = ( ( 𝐴 ↑o 𝐶 ) ·o 𝐴 ) )
15 14 3adant2 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ↑o suc 𝐶 ) = ( ( 𝐴 ↑o 𝐶 ) ·o 𝐴 ) )
16 15 sseq1d ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ↑o suc 𝐶 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ↔ ( ( 𝐴 ↑o 𝐶 ) ·o 𝐴 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ) )
17 13 16 sylibrd ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( 𝐴 ↑o suc 𝐶 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ) )
18 ne0i ⊢ ( 𝐴 ∈ 𝐵 → 𝐵 ≠ ∅ )
19 on0eln0 ⊢ ( 𝐵 ∈ On → ( ∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅ ) )
20 18 19 imbitrrid ⊢ ( 𝐵 ∈ On → ( 𝐴 ∈ 𝐵 → ∅ ∈ 𝐵 ) )
21 20 adantr ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ∅ ∈ 𝐵 ) )
22 oen0 ⊢ ( ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ ∅ ∈ 𝐵 ) → ∅ ∈ ( 𝐵 ↑o 𝐶 ) )
23 22 ex ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ∅ ∈ 𝐵 → ∅ ∈ ( 𝐵 ↑o 𝐶 ) ) )
24 21 23 syld ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ∅ ∈ ( 𝐵 ↑o 𝐶 ) ) )
25 omordi ⊢ ( ( ( 𝐵 ∈ On ∧ ( 𝐵 ↑o 𝐶 ) ∈ On ) ∧ ∅ ∈ ( 𝐵 ↑o 𝐶 ) ) → ( 𝐴 ∈ 𝐵 → ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐵 ) ) )
26 8 25 syldanl ⊢ ( ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ ∅ ∈ ( 𝐵 ↑o 𝐶 ) ) → ( 𝐴 ∈ 𝐵 → ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐵 ) ) )
27 26 ex ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ∅ ∈ ( 𝐵 ↑o 𝐶 ) → ( 𝐴 ∈ 𝐵 → ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐵 ) ) ) )
28 27 com23 ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( ∅ ∈ ( 𝐵 ↑o 𝐶 ) → ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐵 ) ) ) )
29 24 28 mpdd ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐵 ) ) )
30 29 3adant1 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐵 ) ) )
31 oesuc ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 ↑o suc 𝐶 ) = ( ( 𝐵 ↑o 𝐶 ) ·o 𝐵 ) )
32 31 3adant1 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐵 ↑o suc 𝐶 ) = ( ( 𝐵 ↑o 𝐶 ) ·o 𝐵 ) )
33 32 eleq2d ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ↔ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐵 ) ) )
34 30 33 sylibrd ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) )
35 17 34 jcad ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( ( 𝐴 ↑o suc 𝐶 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∧ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) ) )
36 35 3expa ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( ( 𝐴 ↑o suc 𝐶 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∧ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) ) )
37 onsucb ⊢ ( 𝐶 ∈ On ↔ suc 𝐶 ∈ On )
38 oecl ⊢ ( ( 𝐴 ∈ On ∧ suc 𝐶 ∈ On ) → ( 𝐴 ↑o suc 𝐶 ) ∈ On )
39 oecl ⊢ ( ( 𝐵 ∈ On ∧ suc 𝐶 ∈ On ) → ( 𝐵 ↑o suc 𝐶 ) ∈ On )
40 ontr2 ⊢ ( ( ( 𝐴 ↑o suc 𝐶 ) ∈ On ∧ ( 𝐵 ↑o suc 𝐶 ) ∈ On ) → ( ( ( 𝐴 ↑o suc 𝐶 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∧ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) )
41 38 39 40 syl2an ⊢ ( ( ( 𝐴 ∈ On ∧ suc 𝐶 ∈ On ) ∧ ( 𝐵 ∈ On ∧ suc 𝐶 ∈ On ) ) → ( ( ( 𝐴 ↑o suc 𝐶 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∧ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) )
42 41 anandirs ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ suc 𝐶 ∈ On ) → ( ( ( 𝐴 ↑o suc 𝐶 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∧ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) )
43 37 42 sylan2b ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ On ) → ( ( ( 𝐴 ↑o suc 𝐶 ) ⊆ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∧ ( ( 𝐵 ↑o 𝐶 ) ·o 𝐴 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) )
44 36 43 syld ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) )
45 44 exp31 ⊢ ( 𝐴 ∈ On → ( 𝐵 ∈ On → ( 𝐶 ∈ On → ( 𝐴 ∈ 𝐵 → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) ) ) )
46 45 com4l ⊢ ( 𝐵 ∈ On → ( 𝐶 ∈ On → ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ On → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) ) ) )
47 46 imp ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ On → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) ) )
48 3 47 mpdd ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ∈ 𝐵 → ( 𝐴 ↑o suc 𝐶 ) ∈ ( 𝐵 ↑o suc 𝐶 ) ) )