Theorem ixp0 6697

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 3429	. . . 4 |- ( -. E. z z e. B <-> B = (/) )
2	1	rexbii 2473	. . 3 |- ( E. x e. A -. E. z z e. B <-> E. x e. A B = (/) )
3		rexnalim 2455	. . 3 |- ( E. x e. A -. E. z z e. B -> -. A. x e. A E. z z e. B )
4	2, 3	sylbir 134	. 2 |- ( E. x e. A B = (/) -> -. A. x e. A E. z z e. B )
5		ixpm 6696	. . . 4 |- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B )
6	5	con3i 622	. . 3 |- ( -. A. x e. A E. z z e. B -> -. E. f f e. X_ x e. A B )
7		notm0 3429	. . 3 |- ( -. E. f f e. X_ x e. A B <-> X_ x e. A B = (/) )
8	6, 7	sylib 121	. 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 1343   E.wex 1480   e. wcel 2136   A.wral 2444   E.wrex 2445   (/)c0 3409   X_cixp 6664

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-nul 3410  df-ixp 6665

This theorem is referenced by: (None)