Theorem ixp0 4361

Description: The infinite Cartesian product of a family B(x) with an empty member is empty.

Assertion

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

Proof of Theorem ixp0

Step	Hyp	Ref	Expression
1		n0i 2285	. . . . . . . . 9 |- ((f` x) e. B -> -. B = (/))
2	1	con2i 97	. . . . . . . 8 |- (B = (/) -> -. (f` x) e. B)
3	2	r19.22si 1734	. . . . . . 7 |- (E.x e. A B = (/) -> E.x e. A -. (f` x) e. B)
4		rexnal 1654	. . . . . . 7 |- (E.x e. A -. (f` x) e. B <-> -. A.x e. A (f` x) e. B)
5	3, 4	sylib 198	. . . . . 6 |- (E.x e. A B = (/) -> -. A.x e. A (f` x) e. B)
6	5	intnand 693	. . . . 5 |- (E.x e. A B = (/) -> -. (f Fn A /\ A.x e. A (f` x) e. B))
7		noel 2284	. . . . 5 |- -. f e. (/)
8	6, 7	jctir 293	. . . 4 |- (E.x e. A B = (/) -> (-. (f Fn A /\ A.x e. A (f` x) e. B) /\ -. f e. (/)))
9		pm5.21 677	. . . 4 |- ((-. (f Fn A /\ A.x e. A (f` x) e. B) /\ -. f e. (/)) -> ((f Fn A /\ A.x e. A (f` x) e. B) <-> f e. (/)))
10	8, 9	syl 10	. . 3 |- (E.x e. A B = (/) -> ((f Fn A /\ A.x e. A (f` x) e. B) <-> f e. (/)))
11		visset 1813	. . . 4 |- f e. V
12	11	elixp 4350	. . 3 |- (f e. X_x e. A B <-> (f Fn A /\ A.x e. A (f` x) e. B))
13	10, 12	syl5bb 532	. 2 |- (E.x e. A B = (/) -> (f e. X_x e. A B <-> f e. (/)))
14	13	eqrdv 1473	1 |- (E.x e. A B = (/) -> X_x e. A B = (/))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  (/)c0 2280   Fn wfn 3177  ` cfv 3182  X_cixp 4347

This theorem is referenced by:  ac9s 4764

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-ixp 4348

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