Theorem ixp0 7941

Description: The infinite Cartesian product of a family B ( x ) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 9305. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)

Assertion

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

Proof of Theorem ixp0

Step	Hyp	Ref	Expression
1		nne 2798	. . . 4 |- ( -. B =/= (/) <-> B = (/) )
2	1	rexbii 3041	. . 3 |- ( E. x e. A -. B =/= (/) <-> E. x e. A B = (/) )
3		rexnal 2995	. . 3 |- ( E. x e. A -. B =/= (/) <-> -. A. x e. A B =/= (/) )
4	2, 3	bitr3i 266	. 2 |- ( E. x e. A B = (/) <-> -. A. x e. A B =/= (/) )
5		ixpn0 7940	. . 3 |- ( X_ x e. A B =/= (/) -> A. x e. A B =/= (/) )
6	5	necon1bi 2822	. 2 |- ( -. A. x e. A B =/= (/) -> X_ x e. A B = (/) )
7	4, 6	sylbi 207	1 |- ( E. x e. A B = (/) -> X_ x e. A B = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916  df-ixp 7909

This theorem is referenced by:  vonioo  40896  vonicc  40899