Theorem disjen 7464

Description: A stronger form of pwuninel 6790. We can use pwuninel 6790, 2pwuninel 7462 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 6612	. . . . . . . 8 |- ( x e. ( B X. { ~P U. ran A } ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
2	1	ad2antll 723	. . . . . . 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 750	. . . . . . 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 2516	. . . . . 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 5698	. . . . . . 7 |- ( 1st ` x ) e. _V
6		fvex 5698	. . . . . . 7 |- ( 2nd ` x ) e. _V
7	5, 6	opelrn 5067	. . . . . 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 6790	. . . . . 6 |- -. ~P U. ran A e. ran A
10		xp2nd 6606	. . . . . . . . 9 |- ( x e. ( B X. { ~P U. ran A } ) -> ( 2nd ` x ) e. { ~P U. ran A } )
11	10	ad2antll 723	. . . . . . . 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 3899	. . . . . . . 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 2507	. . . . . 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 573	. . . 4 |- ( ( A e. V /\ B e. W ) -> -. ( x e. A /\ x e. ( B X. { ~P U. ran A } ) ) )
17		elin 3536	. . . 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 3669	. 2 |- ( ( A e. V /\ B e. W ) -> ( A i^i ( B X. { ~P U. ran A } ) ) = (/) )
20		simpr 458	. . 3 |- ( ( A e. V /\ B e. W ) -> B e. W )
21		rnexg 6509	. . . . 5 |- ( A e. V -> ran A e. _V )
22	21	adantr 462	. . . 4 |- ( ( A e. V /\ B e. W ) -> ran A e. _V )
23		uniexg 6376	. . . 4 |- ( ran A e. _V -> U. ran A e. _V )
24		pwexg 4473	. . . 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 7392	. . 3 |- ( ( B e. W /\ ~P U. ran A e. _V ) -> ( B X. { ~P U. ran A } ) ~~ B )
27	20, 25, 26	syl2anc 656	. 2 |- ( ( A e. V /\ B e. W ) -> ( B X. { ~P U. ran A } ) ~~ B )
28	19, 27	jca 529	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 1364   e. wcel 1761   _Vcvv 2970   i^i cin 3324   (/)c0 3634   ~Pcpw 3857   {csn 3874   <.cop 3880   U.cuni 4088   class class class wbr 4289   X. cxp 4834   ran crn 4837   ` cfv 5415   1stc1st 6574   2ndc2nd 6575   ~~ cen 7303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-int 4126  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-1st 6576  df-2nd 6577  df-en 7307

This theorem is referenced by:  disjenex  7465  domss2  7466