Theorem ixpprc 6606

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 6591	. . . . 5 |- ( f e. X_ x e. A B -> f Fn A )
2		fndm 5217	. . . . . 6 |- ( f Fn A -> dom f = A )
3		vex 2684	. . . . . . 7 |- f e. _V
4	3	dmex 4800	. . . . . 6 |- dom f e. _V
5	2, 4	eqeltrrdi 2229	. . . . 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 1577	. . 3 |- ( E. f f e. X_ x e. A B -> A e. _V )
8	7	con3i 621	. 2 |- ( -. A e. _V -> -. E. f f e. X_ x e. A B )
9		notm0 3378	. 2 |- ( -. E. f f e. X_ x e. A B <-> X_ x e. A B = (/) )
10	8, 9	sylib 121	1 |- ( -. A e. _V -> X_ x e. A B = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2681   (/)c0 3358   dom cdm 4534   Fn wfn 5113   X_cixp 6585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126  df-ixp 6586

This theorem is referenced by: (None)