Theorem ixp0 6709

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 3435	. . . 4 |- ( -. E. z z e. B <-> B = (/) )
2	1	rexbii 2477	. . 3 |- ( E. x e. A -. E. z z e. B <-> E. x e. A B = (/) )
3		rexnalim 2459	. . 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 6708	. . . 4 |- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B )
6	5	con3i 627	. . 3 |- ( -. A. x e. A E. z z e. B -> -. E. f f e. X_ x e. A B )
7		notm0 3435	. . 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 1348   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449   (/)c0 3414   X_cixp 6676

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-nul 3415  df-ixp 6677

This theorem is referenced by: (None)