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Theorem ac6 8362
 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 8366, 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:   ,,   ,,,   ,   ,
Allowed substitution hints:   (,)   (,)   ()

Proof of Theorem ac6
StepHypRef Expression
1 ac6.1 . 2
2 ac6.2 . . . 4
3 ssrab2 3430 . . . . . 6
43rgenw 2775 . . . . 5
5 iunss 4134 . . . . 5
64, 5mpbir 202 . . . 4
72, 6ssexi 4350 . . 3
8 numth3 8352 . . 3
97, 8ax-mp 8 . 2
10 ac6.3 . . 3
1110ac6num 8361 . 2
121, 9, 11mp3an12 1270 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wex 1551   wceq 1653   wcel 1726  wral 2707  wrex 2708  crab 2711  cvv 2958   wss 3322  ciun 4095   cdm 4880  wf 5452  cfv 5456  ccrd 7824 This theorem is referenced by:  ac6c4  8363  ac6s  8366 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-ac2 8345 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-suc 4589  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-riota 6551  df-recs 6635  df-en 7112  df-card 7828  df-ac 7999
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