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Theorem ac9s 8941
 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 8380). (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 8938 . . 3
3 n0 3732 . . . 4
4 vex 3034 . . . . . 6
54elixp 7547 . . . . 5
65exbii 1726 . . . 4
73, 6bitr2i 258 . . 3
82, 7sylib 201 . 2
9 ixpn0 7572 . 2
108, 9impbii 192 1
 Colors of variables: wff setvar class Syntax hints:   wb 189   wa 376  wex 1671   wcel 1904   wne 2641  wral 2756  cvv 3031  c0 3722   wfn 5584  cfv 5589  cixp 7540 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-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-reg 8125  ax-inf2 8164  ax-ac2 8911 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  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-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  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-isom 5598  df-riota 6270  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-ixp 7541  df-en 7588  df-r1 8253  df-rank 8254  df-card 8391  df-ac 8565 This theorem is referenced by: (None)
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