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Theorem ixpprc 7500
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 3800 . . 3  |-  ( -.  X_ x  e.  A  B  =  (/)  <->  E. f 
f  e.  X_ x  e.  A  B )
2 ixpfn 7485 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f  Fn  A
)
3 fndm 5685 . . . . . 6  |-  ( f  Fn  A  ->  dom  f  =  A )
4 vex 3121 . . . . . . 7  |-  f  e. 
_V
54dmex 6727 . . . . . 6  |-  dom  f  e.  _V
63, 5syl6eqelr 2564 . . . . 5  |-  ( f  Fn  A  ->  A  e.  _V )
72, 6syl 16 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
87exlimiv 1698 . . 3  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A  e.  _V )
91, 8sylbi 195 . 2  |-  ( -.  X_ x  e.  A  B  =  (/)  ->  A  e.  _V )
109con1i 129 1  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3118   (/)c0 3790   dom cdm 5004    Fn wfn 5588   X_cixp 7479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-ixp 7480
This theorem is referenced by:  ixpexg  7503  ixpssmap2g  7508  ixpssmapg  7509  resixpfo  7517
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