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Theorem ixpprc 6620
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 6605 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f  Fn  A
)
2 fndm 5229 . . . . . 6  |-  ( f  Fn  A  ->  dom  f  =  A )
3 vex 2692 . . . . . . 7  |-  f  e. 
_V
43dmex 4812 . . . . . 6  |-  dom  f  e.  _V
52, 4eqeltrrdi 2232 . . . . 5  |-  ( f  Fn  A  ->  A  e.  _V )
61, 5syl 14 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
76exlimiv 1578 . . 3  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A  e.  _V )
87con3i 622 . 2  |-  ( -.  A  e.  _V  ->  -. 
E. f  f  e.  X_ x  e.  A  B )
9 notm0 3387 . 2  |-  ( -. 
E. f  f  e.  X_ x  e.  A  B 
<-> 
X_ x  e.  A  B  =  (/) )
108, 9sylib 121 1  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1332   E.wex 1469    e. wcel 1481   _Vcvv 2689   (/)c0 3367   dom cdm 4546    Fn wfn 5125   X_cixp 6599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138  df-ixp 6600
This theorem is referenced by: (None)
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