Theorem onxpdisj 5089

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 5002. (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 3872	. 2 |- ( ( On i^i ( _V X. _V ) ) = (/) <-> A. x e. On -. x e. ( _V X. _V ) )
2		on0eqel 5001	. . 3 |- ( x e. On -> ( x = (/) \/ (/) e. x ) )
3		0nelxp 5033	. . . . 5 |- -. (/) e. ( _V X. _V )
4		eleq1 2539	. . . . 5 |- ( x = (/) -> ( x e. ( _V X. _V ) <-> (/) e. ( _V X. _V ) ) )
5	3, 4	mtbiri 303	. . . 4 |- ( x = (/) -> -. x e. ( _V X. _V ) )
6		0nelelxp 5034	. . . . 5 |- ( x e. ( _V X. _V ) -> -. (/) e. x )
7	6	con2i 120	. . . 4 |- ( (/) e. x -> -. x e. ( _V X. _V ) )
8	5, 7	jaoi 379	. . 3 |- ( ( x = (/) \/ (/) e. x ) -> -. x e. ( _V X. _V ) )
9	2, 8	syl 16	. 2 |- ( x e. On -> -. x e. ( _V X. _V ) )
10	1, 9	mprgbir 2831	1 |- ( On i^i ( _V X. _V ) ) = (/)

Syntax hints:   -. wn 3   \/ wo 368   = wceq 1379   e. wcel 1767   _Vcvv 3118   i^i cin 3480   (/)c0 3790   Oncon0 4884   X. cxp 5003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-xp 5011

This theorem is referenced by: (None)