Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixpprc Unicode version

Theorem ixpprc 7510
 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:   ,

Proof of Theorem ixpprc
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 neq0 3795 . . 3
2 ixpfn 7495 . . . . 5
3 fndm 5685 . . . . . 6
4 vex 3112 . . . . . . 7
54dmex 6733 . . . . . 6
63, 5syl6eqelr 2554 . . . . 5
72, 6syl 16 . . . 4
87exlimiv 1722 . . 3
91, 8sylbi 195 . 2
109con1i 129 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109   c0 3784  domcdm 5004  Fnwfn 5588  X_cixp 7489 This theorem is referenced by:  ixpexg  7513  ixpssmap2g  7518  ixpssmapg  7519  resixpfo  7527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  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 7490
 Copyright terms: Public domain W3C validator