Theorem ixpprc 6676

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 6661	. . . . 5 |- ( f e. X_ x e. A B -> f Fn A )
2		fndm 5281	. . . . . 6 |- ( f Fn A -> dom f = A )
3		vex 2724	. . . . . . 7 |- f e. _V
4	3	dmex 4864	. . . . . 6 |- dom f e. _V
5	2, 4	eqeltrrdi 2256	. . . . 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 1585	. . 3 |- ( E. f f e. X_ x e. A B -> A e. _V )
8	7	con3i 622	. 2 |- ( -. A e. _V -> -. E. f f e. X_ x e. A B )
9		notm0 3424	. 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 1342   E.wex 1479   e. wcel 2135   _Vcvv 2721   (/)c0 3404   dom cdm 4598   Fn wfn 5177   X_cixp 6655

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-iota 5147  df-fun 5184  df-fn 5185  df-fv 5190  df-ixp 6656

This theorem is referenced by: (None)