Theorem onxpdisj 6485

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 6484. (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 4403	. 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2		on0eqel 6483	. . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3		0nelxp 5689	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
4		eleq1 2848	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
5	3, 4	mtbiri 330	. . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6		0nelelxp 5690	. . . . 5 ⊢ (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
7	6	con2i 140	. . . 4 ⊢ (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V))
8	5, 7	jaoi 871	. . 3 ⊢ ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V))
9	2, 8	syl 18	. 2 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V))
10	1, 9	mprgbir 3083	1 ⊢ (On ∩ (V × V)) = ∅

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∩ cin 3898  ∅c0 4279   × cxp 5653  Oncon0 6357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-ord 6360  df-on 6361

This theorem is used by: (None)