Theorem onxpdisj 5764

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 5763. (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

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disj 3969	. 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2		on0eqel 5762	. . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3		0nelxp 5067	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
4		eleq1 2676	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
5	3, 4	mtbiri 316	. . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6		0nelelxp 5069	. . . . 5 ⊢ (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
7	6	con2i 133	. . . 4 ⊢ (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V))
8	5, 7	jaoi 393	. . 3 ⊢ ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V))
9	2, 8	syl 17	. 2 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V))
10	1, 9	mprgbir 2911	1 ⊢ (On ∩ (V × V)) = ∅

Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ∅c0 3874   × cxp 5036  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-ord 5643  df-on 5644

This theorem is referenced by: (None)