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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 , 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 6849 . . . . . . . 8
21ad2antll 743 . . . . . . 7
3 simprl 772 . . . . . . 7
42, 3eqeltrrd 2550 . . . . . 6
5 fvex 5889 . . . . . . 7
6 fvex 5889 . . . . . . 7
75, 6opelrn 5072 . . . . . 6
84, 7syl 17 . . . . 5
9 pwuninel 7040 . . . . . 6
10 xp2nd 6843 . . . . . . . . 9
1110ad2antll 743 . . . . . . . 8
12 elsni 3985 . . . . . . . 8
1311, 12syl 17 . . . . . . 7
1413eleq1d 2533 . . . . . 6
159, 14mtbiri 310 . . . . 5
168, 15pm2.65da 586 . . . 4
17 elin 3608 . . . 4
1816, 17sylnibr 312 . . 3
1918eq0rdv 3773 . 2
20 simpr 468 . . 3
21 rnexg 6744 . . . . 5
2221adantr 472 . . . 4
23 uniexg 6607 . . . 4
24 pwexg 4585 . . . 4
2522, 23, 243syl 18 . . 3
26 xpsneng 7675 . . 3
2720, 25, 26syl2anc 673 . 2
2819, 27jca 541 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452   wcel 1904  cvv 3031   cin 3389  c0 3722  cpw 3942  csn 3959  cop 3965  cuni 4190   class class class wbr 4395   cxp 4837   crn 4840  cfv 5589  c1st 6810  c2nd 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
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