Theorem ixpn0 7940

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 Mario Carneiro, 22-Jun-2016.)

Assertion

Ref	Expression
ixpn0	|- ( X_ x e. A B =/= (/) -> A. x e. A B =/= (/) )

Proof of Theorem ixpn0

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. 2 |- ( X_ x e. A B =/= (/) <-> E. f f e. X_ x e. A B )
2		df-ixp 7909	. . . . 5 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
3	2	abeq2i 2735	. . . 4 |- ( f e. X_ x e. A B <-> ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) )
4		ne0i 3921	. . . . 5 |- ( ( f ` x ) e. B -> B =/= (/) )
5	4	ralimi 2952	. . . 4 |- ( A. x e. A ( f ` x ) e. B -> A. x e. A B =/= (/) )
6	3, 5	simplbiim 659	. . 3 |- ( f e. X_ x e. A B -> A. x e. A B =/= (/) )
7	6	exlimiv 1858	. 2 |- ( E. f f e. X_ x e. A B -> A. x e. A B =/= (/) )
8	1, 7	sylbi 207	1 |- ( X_ x e. A B =/= (/) -> A. x e. A B =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   (/)c0 3915   Fn wfn 5883   ` cfv 5888   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-v 3202  df-dif 3577  df-nul 3916  df-ixp 7909

This theorem is referenced by:  ixp0  7941  ac9  9305  ac9s  9315