Theorem ixpprc 7929

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.

Step	Hyp	Ref	Expression
1		neq0 3930	. . 3 |- ( -. X_ x e. A B = (/) <-> E. f f e. X_ x e. A B )
2		ixpfn 7914	. . . . 5 |- ( f e. X_ x e. A B -> f Fn A )
3		fndm 5990	. . . . . 6 |- ( f Fn A -> dom f = A )
4		vex 3203	. . . . . . 7 |- f e. _V
5	4	dmex 7099	. . . . . 6 |- dom f e. _V
6	3, 5	syl6eqelr 2710	. . . . 5 |- ( f Fn A -> A e. _V )
7	2, 6	syl 17	. . . 4 |- ( f e. X_ x e. A B -> A e. _V )
8	7	exlimiv 1858	. . 3 |- ( E. f f e. X_ x e. A B -> A e. _V )
9	1, 8	sylbi 207	. 2 |- ( -. X_ x e. A B = (/) -> A e. _V )
10	9	con1i 144	1 |- ( -. A e. _V -> X_ x e. A B = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   (/)c0 3915   dom cdm 5114   Fn wfn 5883   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ixp 7909

This theorem is referenced by:  ixpexg  7932  ixpssmap2g  7937  ixpssmapg  7938  resixpfo  7946