Theorem disjen 7735

Description: A stronger form of pwuninel 7030. We can use pwuninel 7030, 2pwuninel 7733 to create one or two sets disjoint from a given set A, but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set B we can construct a set x that is equinumerous to it and disjoint from A. (Contributed by Mario Carneiro, 7-Feb-2015.)

Assertion

Ref	Expression
disjen	|- ( ( A e. V /\ B e. W ) -> ( ( A i^i ( B X. { ~P U. ran A } ) ) = (/) /\ ( B X. { ~P U. ran A } ) ~~ B ) )

Proof of Theorem disjen

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1st2nd2 6844	. . . . . . . 8 |- ( x e. ( B X. { ~P U. ran A } ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
2	1	ad2antll 733	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
3		simprl 762	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) ) -> x e. A )
4	2, 3	eqeltrrd 2518	. . . . . 6 |- ( ( ( A e. V /\ B e. W ) /\ ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. A )
5		fvex 5891	. . . . . . 7 |- ( 1st ` x ) e. _V
6		fvex 5891	. . . . . . 7 |- ( 2nd ` x ) e. _V
7	5, 6	opelrn 5086	. . . . . 6 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. A -> ( 2nd ` x ) e. ran A )
8	4, 7	syl 17	. . . . 5 |- ( ( ( A e. V /\ B e. W ) /\ ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) ) -> ( 2nd ` x ) e. ran A )
9		pwuninel 7030	. . . . . 6 |- -. ~P U. ran A e. ran A
10		xp2nd 6838	. . . . . . . . 9 |- ( x e. ( B X. { ~P U. ran A } ) -> ( 2nd ` x ) e. { ~P U. ran A } )
11	10	ad2antll 733	. . . . . . . 8 |- ( ( ( A e. V /\ B e. W ) /\ ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) ) -> ( 2nd ` x ) e. { ~P U. ran A } )
12		elsni 4027	. . . . . . . 8 |- ( ( 2nd ` x ) e. { ~P U. ran A } -> ( 2nd ` x ) = ~P U. ran A )
13	11, 12	syl 17	. . . . . . 7 |- ( ( ( A e. V /\ B e. W ) /\ ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) ) -> ( 2nd ` x ) = ~P U. ran A )
14	13	eleq1d 2498	. . . . . 6 |- ( ( ( A e. V /\ B e. W ) /\ ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) ) -> ( ( 2nd ` x ) e. ran A <-> ~P U. ran A e. ran A ) )
15	9, 14	mtbiri 304	. . . . 5 |- ( ( ( A e. V /\ B e. W ) /\ ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) ) -> -. ( 2nd ` x ) e. ran A )
16	8, 15	pm2.65da 578	. . . 4 |- ( ( A e. V /\ B e. W ) -> -. ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) )
17		elin 3655	. . . 4 |- ( x e. ( A i^i ( B X. { ~P U. ran A } ) ) <-> ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) )
18	16, 17	sylnibr 306	. . 3 |- ( ( A e. V /\ B e. W ) -> -. x e. ( A i^i ( B X. { ~P U. ran A } ) ) )
19	18	eq0rdv 3803	. 2 |- ( ( A e. V /\ B e. W ) -> ( A i^i ( B X. { ~P U. ran A } ) ) = (/) )
20		simpr 462	. . 3 |- ( ( A e. V /\ B e. W ) -> B e. W )
21		rnexg 6739	. . . . 5 |- ( A e. V -> ran A e. _V )
22	21	adantr 466	. . . 4 |- ( ( A e. V /\ B e. W ) -> ran A e. _V )
23		uniexg 6602	. . . 4 |- ( ran A e. _V -> U. ran A e. _V )
24		pwexg 4609	. . . 4 |- ( U. ran A e. _V -> ~P U. ran A e. _V )
25	22, 23, 24	3syl 18	. . 3 |- ( ( A e. V /\ B e. W ) -> ~P U. ran A e. _V )
26		xpsneng 7663	. . 3 |- ( ( B e. W /\ ~P U. ran A e. _V ) -> ( B X. { ~P U. ran A } ) ~~ B )
27	20, 25, 26	syl2anc 665	. 2 |- ( ( A e. V /\ B e. W ) -> ( B X. { ~P U. ran A } ) ~~ B )
28	19, 27	jca 534	1 |- ( ( A e. V /\ B e. W ) -> ( ( A i^i ( B X. { ~P U. ran A } ) ) = (/) /\ ( B X. { ~P U. ran A } ) ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1870   _Vcvv 3087   i^i cin 3441   (/)c0 3767   ~Pcpw 3985   {csn 4002   <.cop 4008   U.cuni 4222   class class class wbr 4426   X. cxp 4852   ran crn 4855   ` cfv 5601   1stc1st 6805   2ndc2nd 6806   ~~ cen 7574

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-8 1872  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-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  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-nel 2628  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-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-1st 6807  df-2nd 6808  df-en 7578

This theorem is referenced by:  disjenex  7736  domss2  7737