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Theorem ac9s 8378
 Description: An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. This is a stronger version of the axiom in Enderton, with no existence requirement for the family of classes (achieved via the Collection Principle cp 7820). (Contributed by NM, 29-Sep-2006.)
Hypothesis
Ref Expression
ac9.1
Assertion
Ref Expression
ac9s
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem ac9s
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ac9.1 . . . 4
21ac6s4 8375 . . 3
3 n0 3639 . . . 4
4 vex 2961 . . . . . 6
54elixp 7072 . . . . 5
65exbii 1593 . . . 4
73, 6bitr2i 243 . . 3
82, 7sylib 190 . 2
9 ixpn0 7097 . 2
108, 9impbii 182 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360  wex 1551   wcel 1726   wne 2601  wral 2707  cvv 2958  c0 3630   wfn 5452  cfv 5457  cixp 7066 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-reg 7563  ax-inf2 7599  ax-ac2 8348 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-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-recs 6636  df-rdg 6671  df-ixp 7067  df-en 7113  df-r1 7693  df-rank 7694  df-card 7831  df-ac 8002
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