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 4416	. 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2		on0eqel 6487	. . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3		0nelxp 5696	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
4		eleq1 2857	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
5	3, 4	mtbiri 330	. . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6		0nelelxp 5697	. . . . 5 ⊢ (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
7	6	con2i 140	. . . 4 ⊢ (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V))
8	5, 7	jaoi 870	. . 3 ⊢ ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V))
9	2, 8	syl 18	. 2 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V))
10	1, 9	mprgbir 3092	1 ⊢ (On ∩ (V × V)) = ∅

Syntax hints:  ¬ wn 3   ∨ wo 860   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912  ∅c0 4294   × cxp 5660  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-ord 6364  df-on 6365

This theorem is referenced by: (None)