Theorem ixpn0 7521

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 Mario Carneiro, 22-Jun-2016.)

Assertion

Ref	Expression
ixpn0

Proof of Theorem ixpn0

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3794	. 2
2		df-ixp 7490	. . . . . 6
3	2	abeq2i 2584	. . . . 5
4	3	simprbi 464	. . . 4
5		ne0i 3790	. . . . 5
6	5	ralimi 2850	. . . 4
7	4, 6	syl 16	. . 3
8	7	exlimiv 1722	. 2
9	1, 8	sylbi 195	1

Syntax hints:  ->wi 4  /\wa 369  E.wex 1612  e.wcel 1818  {cab 2442  =/=wne 2652  A.wral 2807  c0 3784  Fnwfn 5588  `cfv 5593  X_cixp 7489

This theorem is referenced by:  ixp0  7522  ac9  8884  ac9s  8894

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-v 3111  df-dif 3478  df-nul 3785  df-ixp 7490