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Theorem ixpprc 7498
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 3715 . . 3  |-  ( -.  X_ x  e.  A  B  =  (/)  <->  E. f 
f  e.  X_ x  e.  A  B )
2 ixpfn 7483 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f  Fn  A
)
3 fndm 5636 . . . . . 6  |-  ( f  Fn  A  ->  dom  f  =  A )
4 vex 3025 . . . . . . 7  |-  f  e. 
_V
54dmex 6684 . . . . . 6  |-  dom  f  e.  _V
63, 5syl6eqelr 2515 . . . . 5  |-  ( f  Fn  A  ->  A  e.  _V )
72, 6syl 17 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
87exlimiv 1770 . . 3  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A  e.  _V )
91, 8sylbi 198 . 2  |-  ( -.  X_ x  e.  A  B  =  (/)  ->  A  e.  _V )
109con1i 132 1  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3022   (/)c0 3704   dom cdm 4796    Fn wfn 5539   X_cixp 7477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552  df-ixp 7478
This theorem is referenced by:  ixpexg  7501  ixpssmap2g  7506  ixpssmapg  7507  resixpfo  7515
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