Theorem onxpdisj 6438

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 6437. (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 4399	. 2 ⊢ ((On ∩ (V × V)) = ∅ ↔ ∀𝑥 ∈ On ¬ 𝑥 ∈ (V × V))
2		on0eqel 6436	. . 3 ⊢ (𝑥 ∈ On → (𝑥 = ∅ ∨ ∅ ∈ 𝑥))
3		0nelxp 5653	. . . . 5 ⊢ ¬ ∅ ∈ (V × V)
4		eleq1 2821	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
5	3, 4	mtbiri 327	. . . 4 ⊢ (𝑥 = ∅ → ¬ 𝑥 ∈ (V × V))
6		0nelelxp 5654	. . . . 5 ⊢ (𝑥 ∈ (V × V) → ¬ ∅ ∈ 𝑥)
7	6	con2i 139	. . . 4 ⊢ (∅ ∈ 𝑥 → ¬ 𝑥 ∈ (V × V))
8	5, 7	jaoi 857	. . 3 ⊢ ((𝑥 = ∅ ∨ ∅ ∈ 𝑥) → ¬ 𝑥 ∈ (V × V))
9	2, 8	syl 17	. 2 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ (V × V))
10	1, 9	mprgbir 3055	1 ⊢ (On ∩ (V × V)) = ∅

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897  ∅c0 4282   × cxp 5617  Oncon0 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-tr 5201  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-ord 6314  df-on 6315

This theorem is referenced by: (None)