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.

Step	Hyp	Ref	Expression
1		neq0 3795	. . 3
2		ixpfn 7495	. . . . 5
3		fndm 5685	. . . . . 6
4		vex 3112	. . . . . . 7
5	4	dmex 6733	. . . . . 6
6	3, 5	syl6eqelr 2554	. . . . 5
7	2, 6	syl 16	. . . 4
8	7	exlimiv 1722	. . 3
9	1, 8	sylbi 195	. 2
10	9	con1i 129	1

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