Theorem ixp0 6841

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 3489	. . . 4 |- ( -. E. z z e. B <-> B = (/) )
2	1	rexbii 2515	. . 3 |- ( E. x e. A -. E. z z e. B <-> E. x e. A B = (/) )
3		rexnalim 2497	. . 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 6840	. . . 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 3489	. . 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 1373   E.wex 1516   e. wcel 2178   A.wral 2486   E.wrex 2487   (/)c0 3468   X_cixp 6808

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-nul 3469  df-ixp 6809

This theorem is referenced by: (None)