Theorem ixp0 6528

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 3322	. . . 4 |- ( -. E. z z e. B <-> B = (/) )
2	1	rexbii 2396	. . 3 |- ( E. x e. A -. E. z z e. B <-> E. x e. A B = (/) )
3		rexnalim 2381	. . 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 6527	. . . 4 |- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B )
6	5	con3i 600	. . 3 |- ( -. A. x e. A E. z z e. B -> -. E. f f e. X_ x e. A B )
7		notm0 3322	. . 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 1296   E.wex 1433   e. wcel 1445   A.wral 2370   E.wrex 2371   (/)c0 3302   X_cixp 6495

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-nul 3303  df-ixp 6496

This theorem is referenced by: (None)