Theorem ixpprc 6721

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		ixpfn 6706	. . . . 5 |- ( f e. X_ x e. A B -> f Fn A )
2		fndm 5317	. . . . . 6 |- ( f Fn A -> dom f = A )
3		vex 2742	. . . . . . 7 |- f e. _V
4	3	dmex 4895	. . . . . 6 |- dom f e. _V
5	2, 4	eqeltrrdi 2269	. . . . 5 |- ( f Fn A -> A e. _V )
6	1, 5	syl 14	. . . 4 |- ( f e. X_ x e. A B -> A e. _V )
7	6	exlimiv 1598	. . 3 |- ( E. f f e. X_ x e. A B -> A e. _V )
8	7	con3i 632	. 2 |- ( -. A e. _V -> -. E. f f e. X_ x e. A B )
9		notm0 3445	. 2 |- ( -. E. f f e. X_ x e. A B <-> X_ x e. A B = (/) )
10	8, 9	sylib 122	1 |- ( -. A e. _V -> X_ x e. A B = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2739   (/)c0 3424   dom cdm 4628   Fn wfn 5213   X_cixp 6700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ixp 6701

This theorem is referenced by: (None)