Theorem ixp0 6799

Description: The infinite Cartesian product of a family B ( x ) with an empty member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim Kingdon, 16-Feb-2023.)

Assertion

Ref	Expression
ixp0	|- ( E. x e. A B = (/) -> X_ x e. A B = (/) )

Proof of Theorem ixp0

Dummy variables f z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		notm0 3472	. . . 4 |- ( -. E. z z e. B <-> B = (/) )
2	1	rexbii 2504	. . 3 |- ( E. x e. A -. E. z z e. B <-> E. x e. A B = (/) )
3		rexnalim 2486	. . 3 |- ( E. x e. A -. E. z z e. B -> -. A. x e. A E. z z e. B )
4	2, 3	sylbir 135	. 2 |- ( E. x e. A B = (/) -> -. A. x e. A E. z z e. B )
5		ixpm 6798	. . . 4 |- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B )
6	5	con3i 633	. . 3 |- ( -. A. x e. A E. z z e. B -> -. E. f f e. X_ x e. A B )
7		notm0 3472	. . 3 |- ( -. E. f f e. X_ x e. A B <-> X_ x e. A B = (/) )
8	6, 7	sylib 122	. 2 |- ( -. A. x e. A E. z z e. B -> X_ x e. A B = (/) )
9	4, 8	syl 14	1 |- ( E. x e. A B = (/) -> X_ x e. A B = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   E.wex 1506   e. wcel 2167   A.wral 2475   E.wrex 2476   (/)c0 3451   X_cixp 6766

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-nul 3452  df-ixp 6767

This theorem is referenced by: (None)