Theorem onxpdisj 5561

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 5560. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
onxpdisj	|- ( On i^i ( _V X. _V ) ) = (/)

Proof of Theorem onxpdisj

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disj 3817	. 2 |- ( ( On i^i ( _V X. _V ) ) = (/) <-> A. x e. On -. x e. ( _V X. _V ) )
2		on0eqel 5559	. . 3 |- ( x e. On -> ( x = (/) \/ (/) e. x ) )
3		0nelxp 4881	. . . . 5 |- -. (/) e. ( _V X. _V )
4		eleq1 2528	. . . . 5 |- ( x = (/) -> ( x e. ( _V X. _V ) <-> (/) e. ( _V X. _V ) ) )
5	3, 4	mtbiri 309	. . . 4 |- ( x = (/) -> -. x e. ( _V X. _V ) )
6		0nelelxp 4882	. . . . 5 |- ( x e. ( _V X. _V ) -> -. (/) e. x )
7	6	con2i 125	. . . 4 |- ( (/) e. x -> -. x e. ( _V X. _V ) )
8	5, 7	jaoi 385	. . 3 |- ( ( x = (/) \/ (/) e. x ) -> -. x e. ( _V X. _V ) )
9	2, 8	syl 17	. 2 |- ( x e. On -> -. x e. ( _V X. _V ) )
10	1, 9	mprgbir 2764	1 |- ( On i^i ( _V X. _V ) ) = (/)

Syntax hints:   -. wn 3   \/ wo 374   = wceq 1455   e. wcel 1898   _Vcvv 3057   i^i cin 3415   (/)c0 3743   X. cxp 4851   Oncon0 5442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-tr 4512  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-ord 5445  df-on 5446

This theorem is referenced by: (None)