Theorem onxpdisj 6489

Description: Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 6488. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
onxpdisj	⊢ (On ∩ (V × V)) = ∅

Proof of Theorem onxpdisj

Step	Hyp	Ref	Expression
1		disj 4443	. 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2		on0eqel 6487	. . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3		0nelxp 5706	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
4		eleq1 2816	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
5	3, 4	mtbiri 327	. . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6		0nelelxp 5707	. . . . 5 ⊢ (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
7	6	con2i 139	. . . 4 ⊢ (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V))
8	5, 7	jaoi 856	. . 3 ⊢ ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V))
9	2, 8	syl 17	. 2 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V))
10	1, 9	mprgbir 3063	1 ⊢ (On ∩ (V × V)) = ∅

Syntax hints:  ¬ wn 3   ∨ wo 846   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ∩ cin 3943  ∅c0 4318   × cxp 5670  Oncon0 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-ord 6366  df-on 6367

This theorem is referenced by: (None)