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Theorem disjen 7694
 Description: A stronger form of pwuninel 7023. We can use pwuninel 7023, 2pwuninel 7692 to create one or two sets disjoint from a given set , but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set we can construct a set that is equinumerous to it and disjoint from . (Contributed by Mario Carneiro, 7-Feb-2015.)
Assertion
Ref Expression
disjen

Proof of Theorem disjen
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 1st2nd2 6837 . . . . . . . 8
21ad2antll 728 . . . . . . 7
3 simprl 756 . . . . . . 7
42, 3eqeltrrd 2546 . . . . . 6
5 fvex 5881 . . . . . . 7
6 fvex 5881 . . . . . . 7
75, 6opelrn 5239 . . . . . 6
84, 7syl 16 . . . . 5
9 pwuninel 7023 . . . . . 6
10 xp2nd 6831 . . . . . . . . 9
1110ad2antll 728 . . . . . . . 8
12 elsni 4054 . . . . . . . 8
1311, 12syl 16 . . . . . . 7
1413eleq1d 2526 . . . . . 6
159, 14mtbiri 303 . . . . 5
168, 15pm2.65da 576 . . . 4
17 elin 3686 . . . 4
1816, 17sylnibr 305 . . 3
1918eq0rdv 3820 . 2
20 simpr 461 . . 3
21 rnexg 6732 . . . . 5
2221adantr 465 . . . 4
23 uniexg 6597 . . . 4
24 pwexg 4636 . . . 4
2522, 23, 243syl 20 . . 3
26 xpsneng 7622 . . 3
2720, 25, 26syl2anc 661 . 2
2819, 27jca 532 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  i^icin 3474   c0 3784  ~Pcpw 4012  {csn 4029  <.cop 4035  U.cuni 4249   class class class wbr 4452  X.cxp 5002  rancrn 5005  `cfv 5593   c1st 6798   c2nd 6799   cen 7533 This theorem is referenced by:  disjenex  7695  domss2  7696 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-int 4287  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-1st 6800  df-2nd 6801  df-en 7537
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