Theorem disjen 7747

Description: A stronger form of pwuninel 7040. We can use pwuninel 7040, 2pwuninel 7745 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 6849	. . . . . . . 8 |- ( x e. ( B X. { ~P U. ran A } ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
2	1	ad2antll 743	. . . . . . 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 772	. . . . . . 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 2550	. . . . . 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 5889	. . . . . . 7 |- ( 1st ` x ) e. _V
6		fvex 5889	. . . . . . 7 |- ( 2nd ` x ) e. _V
7	5, 6	opelrn 5072	. . . . . 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 7040	. . . . . 6 |- -. ~P U. ran A e. ran A
10		xp2nd 6843	. . . . . . . . 9 |- ( x e. ( B X. { ~P U. ran A } ) -> ( 2nd ` x ) e. { ~P U. ran A } )
11	10	ad2antll 743	. . . . . . . 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 3985	. . . . . . . 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 2533	. . . . . 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 310	. . . . 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 586	. . . 4 |- ( ( A e. V /\ B e. W ) -> -. ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) )
17		elin 3608	. . . 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 312	. . 3 |- ( ( A e. V /\ B e. W ) -> -. x e. ( A i^i ( B X. { ~P U. ran A } ) ) )
19	18	eq0rdv 3773	. 2 |- ( ( A e. V /\ B e. W ) -> ( A i^i ( B X. { ~P U. ran A } ) ) = (/) )
20		simpr 468	. . 3 |- ( ( A e. V /\ B e. W ) -> B e. W )
21		rnexg 6744	. . . . 5 |- ( A e. V -> ran A e. _V )
22	21	adantr 472	. . . 4 |- ( ( A e. V /\ B e. W ) -> ran A e. _V )
23		uniexg 6607	. . . 4 |- ( ran A e. _V -> U. ran A e. _V )
24		pwexg 4585	. . . 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 7675	. . 3 |- ( ( B e. W /\ ~P U. ran A e. _V ) -> ( B X. { ~P U. ran A } ) ~~ B )
27	20, 25, 26	syl2anc 673	. 2 |- ( ( A e. V /\ B e. W ) -> ( B X. { ~P U. ran A } ) ~~ B )
28	19, 27	jca 541	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 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   i^i cin 3389   (/)c0 3722   ~Pcpw 3942   {csn 3959   <.cop 3965   U.cuni 4190   class class class wbr 4395   X. cxp 4837   ran crn 4840   ` cfv 5589   1stc1st 6810   2ndc2nd 6811   ~~ cen 7584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-1st 6812  df-2nd 6813  df-en 7588

This theorem is referenced by:  disjenex  7748  domss2  7749