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Metamath Proof Explorer

Theorem rankunb

Description: The rank of the union of two sets. Theorem 15.17(iii) of Monk1 p. 112. (Contributed by Mario Carneiro, 10-Jun-2013) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Assertion rankunb ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 unwf ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) ↔ ( 𝐴 ∪ 𝐵 ) ∈ ∪ ( 𝑅1 “ On ) )
2 rankval3b ⊢ ( ( 𝐴 ∪ 𝐵 ) ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∩ { 𝑦 ∈ On ∣ ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 } )
3 1 2 sylbi ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ∩ { 𝑦 ∈ On ∣ ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 } )
4 3 eleq2d ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( 𝑥 ∈ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ 𝑥 ∈ ∩ { 𝑦 ∈ On ∣ ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 } ) )
5 vex ⊢ 𝑥 ∈ V
6 5 elintrab ⊢ ( 𝑥 ∈ ∩ { 𝑦 ∈ On ∣ ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 } ↔ ∀ 𝑦 ∈ On ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 → 𝑥 ∈ 𝑦 ) )
7 4 6 bitrdi ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( 𝑥 ∈ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ↔ ∀ 𝑦 ∈ On ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 → 𝑥 ∈ 𝑦 ) ) )
8 elun ⊢ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) )
9 rankelb ⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( 𝑥 ∈ 𝐴 → ( rank ‘ 𝑥 ) ∈ ( rank ‘ 𝐴 ) ) )
10 elun1 ⊢ ( ( rank ‘ 𝑥 ) ∈ ( rank ‘ 𝐴 ) → ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) )
11 9 10 syl6 ⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → ( 𝑥 ∈ 𝐴 → ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
12 rankelb ⊢ ( 𝐵 ∈ ∪ ( 𝑅1 “ On ) → ( 𝑥 ∈ 𝐵 → ( rank ‘ 𝑥 ) ∈ ( rank ‘ 𝐵 ) ) )
13 elun2 ⊢ ( ( rank ‘ 𝑥 ) ∈ ( rank ‘ 𝐵 ) → ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) )
14 12 13 syl6 ⊢ ( 𝐵 ∈ ∪ ( 𝑅1 “ On ) → ( 𝑥 ∈ 𝐵 → ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
15 11 14 jaao ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) → ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
16 8 15 biimtrid ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) → ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
17 16 ralrimiv ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) )
18 rankon ⊢ ( rank ‘ 𝐴 ) ∈ On
19 rankon ⊢ ( rank ‘ 𝐵 ) ∈ On
20 18 19 onun2i ⊢ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ∈ On
21 eleq2 ⊢ ( 𝑦 = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) → ( ( rank ‘ 𝑥 ) ∈ 𝑦 ↔ ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
22 21 ralbidv ⊢ ( 𝑦 = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) → ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 ↔ ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
23 eleq2 ⊢ ( 𝑦 = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) → ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
24 22 23 imbi12d ⊢ ( 𝑦 = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) → ( ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 → 𝑥 ∈ 𝑦 ) ↔ ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) → 𝑥 ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) ) )
25 24 rspcv ⊢ ( ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ∈ On → ( ∀ 𝑦 ∈ On ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 → 𝑥 ∈ 𝑦 ) → ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) → 𝑥 ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) ) )
26 20 25 ax-mp ⊢ ( ∀ 𝑦 ∈ On ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 → 𝑥 ∈ 𝑦 ) → ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) → 𝑥 ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
27 17 26 syl5com ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( ∀ 𝑦 ∈ On ( ∀ 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ( rank ‘ 𝑥 ) ∈ 𝑦 → 𝑥 ∈ 𝑦 ) → 𝑥 ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
28 7 27 sylbid ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( 𝑥 ∈ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) → 𝑥 ∈ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ) )
29 28 ssrdv ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ⊆ ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) )
30 ssun1 ⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 )
31 rankssb ⊢ ( ( 𝐴 ∪ 𝐵 ) ∈ ∪ ( 𝑅1 “ On ) → ( 𝐴 ⊆ ( 𝐴 ∪ 𝐵 ) → ( rank ‘ 𝐴 ) ⊆ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
32 30 31 mpi ⊢ ( ( 𝐴 ∪ 𝐵 ) ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ 𝐴 ) ⊆ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) )
33 ssun2 ⊢ 𝐵 ⊆ ( 𝐴 ∪ 𝐵 )
34 rankssb ⊢ ( ( 𝐴 ∪ 𝐵 ) ∈ ∪ ( 𝑅1 “ On ) → ( 𝐵 ⊆ ( 𝐴 ∪ 𝐵 ) → ( rank ‘ 𝐵 ) ⊆ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
35 33 34 mpi ⊢ ( ( 𝐴 ∪ 𝐵 ) ∈ ∪ ( 𝑅1 “ On ) → ( rank ‘ 𝐵 ) ⊆ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) )
36 32 35 unssd ⊢ ( ( 𝐴 ∪ 𝐵 ) ∈ ∪ ( 𝑅1 “ On ) → ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) )
37 1 36 sylbi ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) ⊆ ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) )
38 29 37 eqssd ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ ∪ ( 𝑅1 “ On ) ) → ( rank ‘ ( 𝐴 ∪ 𝐵 ) ) = ( ( rank ‘ 𝐴 ) ∪ ( rank ‘ 𝐵 ) ) )