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Theorem ac9 8105
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. (Contributed by Mario Carneiro, 22-Mar-2013.)
Hypotheses
Ref Expression
ac6c4.1  |-  A  e. 
_V
ac6c4.2  |-  B  e. 
_V
Assertion
Ref Expression
ac9  |-  ( A. x  e.  A  B  =/=  (/)  <->  X_ x  e.  A  B  =/=  (/) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem ac9
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ac6c4.1 . . . 4  |-  A  e. 
_V
2 ac6c4.2 . . . 4  |-  B  e. 
_V
31, 2ac6c4 8103 . . 3  |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
4 n0 3464 . . . 4  |-  ( X_ x  e.  A  B  =/=  (/)  <->  E. f  f  e.  X_ x  e.  A  B )
5 vex 2791 . . . . . 6  |-  f  e. 
_V
65elixp 6818 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
76exbii 1569 . . . 4  |-  ( E. f  f  e.  X_ x  e.  A  B  <->  E. f ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B
) )
84, 7bitr2i 241 . . 3  |-  ( E. f ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B
)  <->  X_ x  e.  A  B  =/=  (/) )
93, 8sylib 188 . 2  |-  ( A. x  e.  A  B  =/=  (/)  ->  X_ x  e.  A  B  =/=  (/) )
10 ixp0 6844 . . . 4  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
1110necon3ai 2486 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  -.  E. x  e.  A  B  =  (/) )
12 df-ne 2448 . . . . 5  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
1312ralbii 2567 . . . 4  |-  ( A. x  e.  A  B  =/=  (/)  <->  A. x  e.  A  -.  B  =  (/) )
14 ralnex 2553 . . . 4  |-  ( A. x  e.  A  -.  B  =  (/)  <->  -.  E. x  e.  A  B  =  (/) )
1513, 14bitri 240 . . 3  |-  ( A. x  e.  A  B  =/=  (/)  <->  -.  E. x  e.  A  B  =  (/) )
1611, 15sylibr 203 . 2  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
179, 16impbii 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 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788   (/)c0 3455    Fn wfn 5215   ` cfv 5220   X_cixp 6812
This theorem is referenced by:  konigthlem  8185
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4186  ax-pr 4212  ax-un 4510  ax-ac2 8084
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4303  df-id 4307  df-po 4312  df-so 4313  df-fr 4350  df-se 4351  df-we 4352  df-ord 4393  df-on 4394  df-suc 4396  df-xp 4693  df-rel 4694  df-cnv 4695  df-co 4696  df-dm 4697  df-rn 4698  df-res 4699  df-ima 4700  df-fun 5222  df-fn 5223  df-f 5224  df-f1 5225  df-fo 5226  df-f1o 5227  df-fv 5228  df-isom 5229  df-iota 6252  df-riota 6299  df-recs 6383  df-ixp 6813  df-en 6859  df-card 7567  df-ac 7738
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