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 3839	. 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 4882	. . . . 5 |- -. (/) e. ( _V X. _V )
4		eleq1 2501	. . . . 5 |- ( x = (/) -> ( x e. ( _V X. _V ) <-> (/) e. ( _V X. _V ) ) )
5	3, 4	mtbiri 304	. . . 4 |- ( x = (/) -> -. x e. ( _V X. _V ) )
6		0nelelxp 4883	. . . . 5 |- ( x e. ( _V X. _V ) -> -. (/) e. x )
7	6	con2i 123	. . . 4 |- ( (/) e. x -> -. x e. ( _V X. _V ) )
8	5, 7	jaoi 380	. . 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 2796	1 |- ( On i^i ( _V X. _V ) ) = (/)

Syntax hints:   -. wn 3   \/ wo 369   = wceq 1437   e. wcel 1870   _Vcvv 3087   i^i cin 3441   (/)c0 3767   X. cxp 4852   Oncon0 5442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-ord 5445  df-on 5446

This theorem is referenced by: (None)