Theorem ixpprc 6866

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 6851	. . . . 5 |- ( f e. X_ x e. A B -> f Fn A )
2		fndm 5420	. . . . . 6 |- ( f Fn A -> dom f = A )
3		vex 2802	. . . . . . 7 |- f e. _V
4	3	dmex 4991	. . . . . 6 |- dom f e. _V
5	2, 4	eqeltrrdi 2321	. . . . 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 1644	. . 3 |- ( E. f f e. X_ x e. A B -> A e. _V )
8	7	con3i 635	. 2 |- ( -. A e. _V -> -. E. f f e. X_ x e. A B )
9		notm0 3512	. 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 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   (/)c0 3491   dom cdm 4719   Fn wfn 5313   X_cixp 6845

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ixp 6846

This theorem is referenced by: (None)