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Theorem ixpprc 7019
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 3581 . . 3  |-  ( -.  X_ x  e.  A  B  =  (/)  <->  E. f 
f  e.  X_ x  e.  A  B )
2 ixpfn 7004 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f  Fn  A
)
3 fndm 5484 . . . . . 6  |-  ( f  Fn  A  ->  dom  f  =  A )
4 vex 2902 . . . . . . 7  |-  f  e. 
_V
54dmex 5072 . . . . . 6  |-  dom  f  e.  _V
63, 5syl6eqelr 2476 . . . . 5  |-  ( f  Fn  A  ->  A  e.  _V )
72, 6syl 16 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
87exlimiv 1641 . . 3  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A  e.  _V )
91, 8sylbi 188 . 2  |-  ( -.  X_ x  e.  A  B  =  (/)  ->  A  e.  _V )
109con1i 123 1  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2899   (/)c0 3571   dom cdm 4818    Fn wfn 5389   X_cixp 6999
This theorem is referenced by:  ixpexg  7022  ixpssmap2g  7027  ixpssmapg  7028  resixpfo  7036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402  df-ixp 7000
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