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Theorem ac6 8881
Description: Equivalent of Axiom of Choice. This is useful for proving that there exists, for example, a sequence mapping natural numbers to members of a larger set , where depends on (the natural number) and (to specify a member of ). A stronger version of this theorem, ac6s 8885, allows to be a proper class. (Contributed by NM, 18-Oct-1999.) (Revised by Mario Carneiro, 27-Aug-2015.)
Hypotheses
Ref Expression
ac6.1
ac6.2
ac6.3
Assertion
Ref Expression
ac6
Distinct variable groups:   , ,   , , ,   ,   ,

Proof of Theorem ac6
StepHypRef Expression
1 ac6.1 . 2
2 ac6.2 . . . 4
3 ssrab2 3584 . . . . . 6
43rgenw 2818 . . . . 5
5 iunss 4371 . . . . 5
64, 5mpbir 209 . . . 4
72, 6ssexi 4597 . . 3
8 numth3 8871 . . 3
97, 8ax-mp 5 . 2
10 ac6.3 . . 3
1110ac6num 8880 . 2
121, 9, 11mp3an12 1314 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808  {crab 2811   cvv 3109  C_wss 3475  U_ciun 4330  domcdm 5004  -->wf 5589  `cfv 5593   ccrd 8337
This theorem is referenced by:  ac6c4  8882  ac6s  8885
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-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-ac2 8864
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-int 4287  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  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-isom 5602  df-riota 6257  df-recs 7061  df-en 7537  df-card 8341  df-ac 8518
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