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Theorem ac9s 8116
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  B ( x ) (achieved via the Collection Principle cp 7557). (Contributed by NM, 29-Sep-2006.)
Hypothesis
Ref Expression
ac9.1  |-  A  e. 
_V
Assertion
Ref Expression
ac9s  |-  ( A. x  e.  A  B  =/=  (/)  <->  X_ x  e.  A  B  =/=  (/) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem ac9s
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ac9.1 . . . 4  |-  A  e. 
_V
21ac6s4 8113 . . 3  |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
3 n0 3465 . . . 4  |-  ( X_ x  e.  A  B  =/=  (/)  <->  E. f  f  e.  X_ x  e.  A  B )
4 vex 2792 . . . . . 6  |-  f  e. 
_V
54elixp 6819 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
65exbii 1569 . . . 4  |-  ( E. f  f  e.  X_ x  e.  A  B  <->  E. f ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B
) )
73, 6bitr2i 241 . . 3  |-  ( E. f ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B
)  <->  X_ x  e.  A  B  =/=  (/) )
82, 7sylib 188 . 2  |-  ( A. x  e.  A  B  =/=  (/)  ->  X_ x  e.  A  B  =/=  (/) )
9 ixp0 6845 . . . 4  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
109necon3ai 2487 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  -.  E. x  e.  A  B  =  (/) )
11 df-ne 2449 . . . . 5  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
1211ralbii 2568 . . . 4  |-  ( A. x  e.  A  B  =/=  (/)  <->  A. x  e.  A  -.  B  =  (/) )
13 ralnex 2554 . . . 4  |-  ( A. x  e.  A  -.  B  =  (/)  <->  -.  E. x  e.  A  B  =  (/) )
1412, 13bitri 240 . . 3  |-  ( A. x  e.  A  B  =/=  (/)  <->  -.  E. x  e.  A  B  =  (/) )
1510, 14sylibr 203 . 2  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
168, 15impbii 180 1  |-  ( A. x  e.  A  B  =/=  (/)  <->  X_ x  e.  A  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   _Vcvv 2789   (/)c0 3456    Fn wfn 5216   ` cfv 5221   X_cixp 6813
This theorem is referenced by:  prl  24578  dstr  24582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-reg 7302  ax-inf2 7338  ax-ac2 8085
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-ixp 6814  df-en 6860  df-r1 7432  df-rank 7433  df-card 7568  df-ac 7739
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