Theorem ixpprc 6516

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 6501	. . . . 5 |- ( f e. X_ x e. A B -> f Fn A )
2		fndm 5147	. . . . . 6 |- ( f Fn A -> dom f = A )
3		vex 2636	. . . . . . 7 |- f e. _V
4	3	dmex 4731	. . . . . 6 |- dom f e. _V
5	2, 4	syl6eqelr 2186	. . . . 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 1541	. . 3 |- ( E. f f e. X_ x e. A B -> A e. _V )
8	7	con3i 600	. 2 |- ( -. A e. _V -> -. E. f f e. X_ x e. A B )
9		notm0 3322	. 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 1296   E.wex 1433   e. wcel 1445   _Vcvv 2633   (/)c0 3302   dom cdm 4467   Fn wfn 5044   X_cixp 6495

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-fv 5057  df-ixp 6496

This theorem is referenced by: (None)