Theorem ixp0 7522

Description: The infinite Cartesian product of a family (x) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 8884. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.)

Assertion

Ref	Expression
ixp0

Proof of Theorem ixp0

Step	Hyp	Ref	Expression
1		nne 2658	. . . 4
2	1	rexbii 2959	. . 3
3		rexnal 2905	. . 3
4	2, 3	bitr3i 251	. 2
5		ixpn0 7521	. . 3
6	5	necon1bi 2690	. 2
7	4, 6	sylbi 195	1

Syntax hints:  -.wn 3  ->wi 4  =wceq 1395  =/=wne 2652  A.wral 2807  E.wrex 2808  c0 3784  X_cixp 7489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-nul 3785  df-ixp 7490