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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 , which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.)
Assertion
Ref Expression
ixpprc
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem ixpprc
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 neq0 3800 . . 3
2 ixpfn 7485 . . . . 5
3 fndm 5685 . . . . . 6
4 vex 3121 . . . . . . 7
54dmex 6727 . . . . . 6
63, 5syl6eqelr 2564 . . . . 5
72, 6syl 16 . . . 4
87exlimiv 1698 . . 3
91, 8sylbi 195 . 2
109con1i 129 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wceq 1379  wex 1596   wcel 1767  cvv 3118  c0 3790   cdm 5004   wfn 5588  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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