Theorem disjen 7674

Description: A stronger form of pwuninel 7004. We can use pwuninel 7004, 2pwuninel 7672 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 6821	. . . . . . . 8 |- ( x e. ( B X. { ~P U. ran A } ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
2	1	ad2antll 728	. . . . . . 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 755	. . . . . . 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 2556	. . . . . 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 5876	. . . . . . 7 |- ( 1st ` x ) e. _V
6		fvex 5876	. . . . . . 7 |- ( 2nd ` x ) e. _V
7	5, 6	opelrn 5234	. . . . . 6 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. A -> ( 2nd ` x ) e. ran A )
8	4, 7	syl 16	. . . . 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 7004	. . . . . 6 |- -. ~P U. ran A e. ran A
10		xp2nd 6815	. . . . . . . . 9 |- ( x e. ( B X. { ~P U. ran A } ) -> ( 2nd ` x ) e. { ~P U. ran A } )
11	10	ad2antll 728	. . . . . . . 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 4052	. . . . . . . 8 |- ( ( 2nd ` x ) e. { ~P U. ran A } -> ( 2nd ` x ) = ~P U. ran A )
13	11, 12	syl 16	. . . . . . 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 2536	. . . . . 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 303	. . . . 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 576	. . . 4 |- ( ( A e. V /\ B e. W ) -> -. ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) )
17		elin 3687	. . . 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 305	. . 3 |- ( ( A e. V /\ B e. W ) -> -. x e. ( A i^i ( B X. { ~P U. ran A } ) ) )
19	18	eq0rdv 3820	. 2 |- ( ( A e. V /\ B e. W ) -> ( A i^i ( B X. { ~P U. ran A } ) ) = (/) )
20		simpr 461	. . 3 |- ( ( A e. V /\ B e. W ) -> B e. W )
21		rnexg 6716	. . . . 5 |- ( A e. V -> ran A e. _V )
22	21	adantr 465	. . . 4 |- ( ( A e. V /\ B e. W ) -> ran A e. _V )
23		uniexg 6581	. . . 4 |- ( ran A e. _V -> U. ran A e. _V )
24		pwexg 4631	. . . 4 |- ( U. ran A e. _V -> ~P U. ran A e. _V )
25	22, 23, 24	3syl 20	. . 3 |- ( ( A e. V /\ B e. W ) -> ~P U. ran A e. _V )
26		xpsneng 7602	. . 3 |- ( ( B e. W /\ ~P U. ran A e. _V ) -> ( B X. { ~P U. ran A } ) ~~ B )
27	20, 25, 26	syl2anc 661	. 2 |- ( ( A e. V /\ B e. W ) -> ( B X. { ~P U. ran A } ) ~~ B )
28	19, 27	jca 532	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 369   = wceq 1379   e. wcel 1767   _Vcvv 3113   i^i cin 3475   (/)c0 3785   ~Pcpw 4010   {csn 4027   <.cop 4033   U.cuni 4245   class class class wbr 4447   X. cxp 4997   ran crn 5000   ` cfv 5588   1stc1st 6782   2ndc2nd 6783   ~~ cen 7513

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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-1st 6784  df-2nd 6785  df-en 7517

This theorem is referenced by:  disjenex  7675  domss2  7676