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Metamath Proof Explorer

Theorem oa00

Description: An ordinal sum is zero iff both of its arguments are zero. Lemma 3.10 of Schloeder p. 8. (Contributed by NM, 6-Dec-2004)

Ref Expression
Assertion oa00 ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o 𝐵 ) = ∅ ↔ ( 𝐴 = ∅ ∧ 𝐵 = ∅ ) ) )

Proof

Step Hyp Ref Expression
1 on0eln0 ⊢ ( 𝐴 ∈ On → ( ∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅ ) )
2 1 adantr ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅ ) )
3 oaword1 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐴 ⊆ ( 𝐴 +o 𝐵 ) )
4 3 sseld ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐴 → ∅ ∈ ( 𝐴 +o 𝐵 ) ) )
5 2 4 sylbird ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ≠ ∅ → ∅ ∈ ( 𝐴 +o 𝐵 ) ) )
6 ne0i ⊢ ( ∅ ∈ ( 𝐴 +o 𝐵 ) → ( 𝐴 +o 𝐵 ) ≠ ∅ )
7 5 6 syl6 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ≠ ∅ → ( 𝐴 +o 𝐵 ) ≠ ∅ ) )
8 7 necon4d ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o 𝐵 ) = ∅ → 𝐴 = ∅ ) )
9 on0eln0 ⊢ ( 𝐵 ∈ On → ( ∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅ ) )
10 9 adantl ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅ ) )
11 0elon ⊢ ∅ ∈ On
12 oaord ⊢ ( ( ∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ∈ 𝐵 ↔ ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) )
13 11 12 mp3an1 ⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ∅ ∈ 𝐵 ↔ ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) )
14 13 ancoms ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ∅ ∈ 𝐵 ↔ ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) )
15 10 14 bitr3d ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 ≠ ∅ ↔ ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) ) )
16 ne0i ⊢ ( ( 𝐴 +o ∅ ) ∈ ( 𝐴 +o 𝐵 ) → ( 𝐴 +o 𝐵 ) ≠ ∅ )
17 15 16 biimtrdi ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐵 ≠ ∅ → ( 𝐴 +o 𝐵 ) ≠ ∅ ) )
18 17 necon4d ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o 𝐵 ) = ∅ → 𝐵 = ∅ ) )
19 8 18 jcad ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o 𝐵 ) = ∅ → ( 𝐴 = ∅ ∧ 𝐵 = ∅ ) ) )
20 oveq12 ⊢ ( ( 𝐴 = ∅ ∧ 𝐵 = ∅ ) → ( 𝐴 +o 𝐵 ) = ( ∅ +o ∅ ) )
21 oa0 ⊢ ( ∅ ∈ On → ( ∅ +o ∅ ) = ∅ )
22 11 21 ax-mp ⊢ ( ∅ +o ∅ ) = ∅
23 20 22 eqtrdi ⊢ ( ( 𝐴 = ∅ ∧ 𝐵 = ∅ ) → ( 𝐴 +o 𝐵 ) = ∅ )
24 19 23 impbid1 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝐴 +o 𝐵 ) = ∅ ↔ ( 𝐴 = ∅ ∧ 𝐵 = ∅ ) ) )