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Theorem ac9s 8122
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 7563). (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 8119 . . 3  |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
3 n0 3466 . . . 4  |-  ( X_ x  e.  A  B  =/=  (/)  <->  E. f  f  e.  X_ x  e.  A  B )
4 vex 2793 . . . . . 6  |-  f  e. 
_V
54elixp 6825 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
65exbii 1571 . . . 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 6851 . . . 4  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
109necon3ai 2488 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  -.  E. x  e.  A  B  =  (/) )
11 df-ne 2450 . . . . 5  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
1211ralbii 2569 . . . 4  |-  ( A. x  e.  A  B  =/=  (/)  <->  A. x  e.  A  -.  B  =  (/) )
13 ralnex 2555 . . . 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 1530    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   E.wrex 2546   _Vcvv 2790   (/)c0 3457    Fn wfn 5252   ` cfv 5257   X_cixp 6819
This theorem is referenced by:  prl  25178  dstr  25182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-reg 7308  ax-inf2 7344  ax-ac2 8091
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-riota 6306  df-recs 6390  df-rdg 6425  df-ixp 6820  df-en 6866  df-r1 7438  df-rank 7439  df-card 7574  df-ac 7745
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