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Theorem ac9s 8136
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 7577). (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 8133 . . 3  |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
3 n0 3477 . . . 4  |-  ( X_ x  e.  A  B  =/=  (/)  <->  E. f  f  e.  X_ x  e.  A  B )
4 vex 2804 . . . . . 6  |-  f  e. 
_V
54elixp 6839 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
65exbii 1572 . . . 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 6865 . . . 4  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
109necon3ai 2499 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  -.  E. x  e.  A  B  =  (/) )
11 df-ne 2461 . . . . 5  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
1211ralbii 2580 . . . 4  |-  ( A. x  e.  A  B  =/=  (/)  <->  A. x  e.  A  -.  B  =  (/) )
13 ralnex 2566 . . . 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801   (/)c0 3468    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  prl  25270  dstr  25274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358  ax-ac2 8105
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-rdg 6439  df-ixp 6834  df-en 6880  df-r1 7452  df-rank 7453  df-card 7588  df-ac 7759
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