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Theorem disjen 7264
 Description: A stronger form of pwuninel 6545. We can use pwuninel 6545, 2pwuninel 7262 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 6386 . . . . . . . 8
21ad2antll 710 . . . . . . 7
3 simprl 733 . . . . . . 7
42, 3eqeltrrd 2511 . . . . . 6
5 fvex 5742 . . . . . . 7
6 fvex 5742 . . . . . . 7
75, 6opelrn 5101 . . . . . 6
84, 7syl 16 . . . . 5
9 pwuninel 6545 . . . . . 6
10 xp2nd 6377 . . . . . . . . 9
1110ad2antll 710 . . . . . . . 8
12 elsni 3838 . . . . . . . 8
1311, 12syl 16 . . . . . . 7
1413eleq1d 2502 . . . . . 6
159, 14mtbiri 295 . . . . 5
168, 15pm2.65da 560 . . . 4
17 elin 3530 . . . 4
1816, 17sylnibr 297 . . 3
1918eq0rdv 3662 . 2
20 simpr 448 . . 3
21 rnexg 5131 . . . . 5
2221adantr 452 . . . 4
23 uniexg 4706 . . . 4
24 pwexg 4383 . . . 4
2522, 23, 243syl 19 . . 3
26 xpsneng 7193 . . 3
2720, 25, 26syl2anc 643 . 2
2819, 27jca 519 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cvv 2956   cin 3319  c0 3628  cpw 3799  csn 3814  cop 3817  cuni 4015   class class class wbr 4212   cxp 4876   crn 4879  cfv 5454  c1st 6347  c2nd 6348   cen 7106 This theorem is referenced by:  disjenex  7265  domss2  7266 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1st 6349  df-2nd 6350  df-en 7110
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