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Theorem ixpprc 6749
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 ixpfn 6734 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f  Fn  A
)
2 fndm 5337 . . . . . 6  |-  ( f  Fn  A  ->  dom  f  =  A )
3 vex 2755 . . . . . . 7  |-  f  e. 
_V
43dmex 4914 . . . . . 6  |-  dom  f  e.  _V
52, 4eqeltrrdi 2281 . . . . 5  |-  ( f  Fn  A  ->  A  e.  _V )
61, 5syl 14 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
76exlimiv 1609 . . 3  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A  e.  _V )
87con3i 633 . 2  |-  ( -.  A  e.  _V  ->  -. 
E. f  f  e.  X_ x  e.  A  B )
9 notm0 3458 . 2  |-  ( -. 
E. f  f  e.  X_ x  e.  A  B 
<-> 
X_ x  e.  A  B  =  (/) )
108, 9sylib 122 1  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1364   E.wex 1503    e. wcel 2160   _Vcvv 2752   (/)c0 3437   dom cdm 4647    Fn wfn 5233   X_cixp 6728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246  df-ixp 6729
This theorem is referenced by: (None)
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