MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ac9s Unicode version

Theorem ac9s 8089
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 7530). (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
StepHypRef Expression
1 ac9.1 . . . 4  |-  A  e. 
_V
21ac6s4 8086 . . 3  |-  ( A. x  e.  A  B  =/=  (/)  ->  E. f
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
3 n0 3439 . . . 4  |-  ( X_ x  e.  A  B  =/=  (/)  <->  E. f  f  e.  X_ x  e.  A  B )
4 vex 2766 . . . . . 6  |-  f  e. 
_V
54elixp 6792 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
65exbii 1580 . . . 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 243 . . 3  |-  ( E. f ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B
)  <->  X_ x  e.  A  B  =/=  (/) )
82, 7sylib 190 . 2  |-  ( A. x  e.  A  B  =/=  (/)  ->  X_ x  e.  A  B  =/=  (/) )
9 ixp0 6818 . . . 4  |-  ( E. x  e.  A  B  =  (/)  ->  X_ x  e.  A  B  =  (/) )
109necon3ai 2461 . . 3  |-  ( X_ x  e.  A  B  =/=  (/)  ->  -.  E. x  e.  A  B  =  (/) )
11 df-ne 2423 . . . . 5  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
1211ralbii 2542 . . . 4  |-  ( A. x  e.  A  B  =/=  (/)  <->  A. x  e.  A  -.  B  =  (/) )
13 ralnex 2528 . . . 4  |-  ( A. x  e.  A  -.  B  =  (/)  <->  -.  E. x  e.  A  B  =  (/) )
1412, 13bitri 242 . . 3  |-  ( A. x  e.  A  B  =/=  (/)  <->  -.  E. x  e.  A  B  =  (/) )
1510, 14sylibr 205 . 2  |-  ( X_ x  e.  A  B  =/=  (/)  ->  A. x  e.  A  B  =/=  (/) )
168, 15impbii 182 1  |-  ( A. x  e.  A  B  =/=  (/)  <->  X_ x  e.  A  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   E.wrex 2519   _Vcvv 2763   (/)c0 3430    Fn wfn 5190   ` cfv 5195   X_cixp 6786
This theorem is referenced by:  prl  24536  dstr  24540
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4106  ax-sep 4116  ax-nul 4124  ax-pow 4161  ax-pr 4187  ax-un 4485  ax-reg 7275  ax-inf2 7311  ax-ac2 8058
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3541  df-pw 3602  df-sn 3621  df-pr 3622  df-tp 3623  df-op 3624  df-uni 3803  df-int 3838  df-iun 3882  df-iin 3883  df-br 3999  df-opab 4053  df-mpt 4054  df-tr 4089  df-eprel 4278  df-id 4282  df-po 4287  df-so 4288  df-fr 4325  df-se 4326  df-we 4327  df-ord 4368  df-on 4369  df-lim 4370  df-suc 4371  df-om 4630  df-xp 4668  df-rel 4669  df-cnv 4670  df-co 4671  df-dm 4672  df-rn 4673  df-res 4674  df-ima 4675  df-fun 5197  df-fn 5198  df-f 5199  df-f1 5200  df-fo 5201  df-f1o 5202  df-fv 5203  df-isom 5204  df-iota 6226  df-riota 6273  df-recs 6357  df-rdg 6392  df-ixp 6787  df-en 6833  df-r1 7405  df-rank 7406  df-card 7541  df-ac 7712
  Copyright terms: Public domain W3C validator