Theorem disjen 7729

Description: A stronger form of pwuninel 7022. We can use pwuninel 7022, 2pwuninel 7727 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 6830	. . . . . . . 8 |- ( x e. ( B X. { ~P U. ran A } ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
2	1	ad2antll 735	. . . . . . 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 764	. . . . . . 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 2530	. . . . . 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 5875	. . . . . . 7 |- ( 1st ` x ) e. _V
6		fvex 5875	. . . . . . 7 |- ( 2nd ` x ) e. _V
7	5, 6	opelrn 5066	. . . . . 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 7022	. . . . . 6 |- -. ~P U. ran A e. ran A
10		xp2nd 6824	. . . . . . . . 9 |- ( x e. ( B X. { ~P U. ran A } ) -> ( 2nd ` x ) e. { ~P U. ran A } )
11	10	ad2antll 735	. . . . . . . 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 3993	. . . . . . . 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 2513	. . . . . 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 305	. . . . 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 580	. . . 4 |- ( ( A e. V /\ B e. W ) -> -. ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) )
17		elin 3617	. . . 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 307	. . 3 |- ( ( A e. V /\ B e. W ) -> -. x e. ( A i^i ( B X. { ~P U. ran A } ) ) )
19	18	eq0rdv 3769	. 2 |- ( ( A e. V /\ B e. W ) -> ( A i^i ( B X. { ~P U. ran A } ) ) = (/) )
20		simpr 463	. . 3 |- ( ( A e. V /\ B e. W ) -> B e. W )
21		rnexg 6725	. . . . 5 |- ( A e. V -> ran A e. _V )
22	21	adantr 467	. . . 4 |- ( ( A e. V /\ B e. W ) -> ran A e. _V )
23		uniexg 6588	. . . 4 |- ( ran A e. _V -> U. ran A e. _V )
24		pwexg 4587	. . . 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 7657	. . 3 |- ( ( B e. W /\ ~P U. ran A e. _V ) -> ( B X. { ~P U. ran A } ) ~~ B )
27	20, 25, 26	syl2anc 667	. 2 |- ( ( A e. V /\ B e. W ) -> ( B X. { ~P U. ran A } ) ~~ B )
28	19, 27	jca 535	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 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   i^i cin 3403   (/)c0 3731   ~Pcpw 3951   {csn 3968   <.cop 3974   U.cuni 4198   class class class wbr 4402   X. cxp 4832   ran crn 4835   ` cfv 5582   1stc1st 6791   2ndc2nd 6792   ~~ cen 7566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-1st 6793  df-2nd 6794  df-en 7570

This theorem is referenced by:  disjenex  7730  domss2  7731