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Theorem ixpprc 6837
Description: A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain  A, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.)
Assertion
Ref Expression
ixpprc  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem ixpprc
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 neq0 3465 . . 3  |-  ( -.  X_ x  e.  A  B  =  (/)  <->  E. f 
f  e.  X_ x  e.  A  B )
2 ixpfn 6822 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f  Fn  A
)
3 fndm 5343 . . . . . 6  |-  ( f  Fn  A  ->  dom  f  =  A )
4 vex 2791 . . . . . . 7  |-  f  e. 
_V
54dmex 4941 . . . . . 6  |-  dom  f  e.  _V
63, 5syl6eqelr 2372 . . . . 5  |-  ( f  Fn  A  ->  A  e.  _V )
72, 6syl 15 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
87exlimiv 1666 . . 3  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A  e.  _V )
91, 8sylbi 187 . 2  |-  ( -.  X_ x  e.  A  B  =  (/)  ->  A  e.  _V )
109con1i 121 1  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   dom cdm 4689    Fn wfn 5250   X_cixp 6817
This theorem is referenced by:  ixpexg  6840  ixpssmap2g  6845  ixpssmapg  6846  resixpfo  6854
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-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ixp 6818
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