Theorem disjenex 8118

Description: Existence version of disjen 8117. (Contributed by Mario Carneiro, 7-Feb-2015.)

Assertion

Ref	Expression
disjenex	|- ( ( A e. V /\ B e. W ) -> E. x ( ( A i^i x ) = (/) /\ x ~~ B ) )

Distinct variable groups:   x, A   x, B   x, V   x, W

Proof of Theorem disjenex

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( A e. V /\ B e. W ) -> B e. W )
2		snex 4908	. . 3 |- { ~P U. ran A } e. _V
3		xpexg 6960	. . 3 |- ( ( B e. W /\ { ~P U. ran A } e. _V ) -> ( B X. { ~P U. ran A } ) e. _V )
4	1, 2, 3	sylancl 694	. 2 |- ( ( A e. V /\ B e. W ) -> ( B X. { ~P U. ran A } ) e. _V )
5		disjen 8117	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A i^i ( B X. { ~P U. ran A } ) ) = (/) /\ ( B X. { ~P U. ran A } ) ~~ B ) )
6		ineq2 3808	. . . . 5 |- ( x = ( B X. { ~P U. ran A } ) -> ( A i^i x ) = ( A i^i ( B X. { ~P U. ran A } ) ) )
7	6	eqeq1d 2624	. . . 4 |- ( x = ( B X. { ~P U. ran A } ) -> ( ( A i^i x ) = (/) <-> ( A i^i ( B X. { ~P U. ran A } ) ) = (/) ) )
8		breq1 4656	. . . 4 |- ( x = ( B X. { ~P U. ran A } ) -> ( x ~~ B <-> ( B X. { ~P U. ran A } ) ~~ B ) )
9	7, 8	anbi12d 747	. . 3 |- ( x = ( B X. { ~P U. ran A } ) -> ( ( ( A i^i x ) = (/) /\ x ~~ B ) <-> ( ( A i^i ( B X. { ~P U. ran A } ) ) = (/) /\ ( B X. { ~P U. ran A } ) ~~ B ) ) )
10	9	spcegv 3294	. 2 |- ( ( B X. { ~P U. ran A } ) e. _V -> ( ( ( A i^i ( B X. { ~P U. ran A } ) ) = (/) /\ ( B X. { ~P U. ran A } ) ~~ B ) -> E. x ( ( A i^i x ) = (/) /\ x ~~ B ) ) )
11	4, 5, 10	sylc 65	1 |- ( ( A e. V /\ B e. W ) -> E. x ( ( A i^i x ) = (/) /\ x ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   X. cxp 5112   ran crn 5115   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169  df-en 7956

This theorem is referenced by: (None)