Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ac6 Structured version   Unicode version

Theorem ac6 8872
 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 8876, 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 3590 . . . . . 6
43rgenw 2828 . . . . 5
5 iunss 4372 . . . . 5
64, 5mpbir 209 . . . 4
72, 6ssexi 4598 . . 3
8 numth3 8862 . . 3
97, 8ax-mp 5 . 2
10 ac6.3 . . 3
1110ac6num 8871 . 2
121, 9, 11mp3an12 1314 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379  wex 1596   wcel 1767  wral 2817  wrex 2818  crab 2821  cvv 3118   wss 3481  ciun 4331   cdm 5005  wf 5590  cfv 5594  ccrd 8328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-ac2 8855 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-recs 7054  df-en 7529  df-card 8332  df-ac 8509 This theorem is referenced by:  ac6c4  8873  ac6s  8876
 Copyright terms: Public domain W3C validator