Theorem ixp0 4345

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 2275	. . . . . . . . 9 |- ((f` x) e. B -> -. B = (/))
2	1	con2i 97	. . . . . . . 8 |- (B = (/) -> -. (f` x) e. B)
3	2	r19.22si 1726	. . . . . . 7 |- (E.x e. A B = (/) -> E.x e. A -. (f` x) e. B)
4		rexnal 1646	. . . . . . 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 691	. . . . 5 |- (E.x e. A B = (/) -> -. (f Fn A /\ A.x e. A (f` x) e. B))
7		noel 2274	. . . . 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		pm5.21 675	. . . 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 1804	. . . 4 |- f e. V
12	11	elixp 4334	. . 3 |- (f e. X_x e. A B <-> (f Fn A /\ A.x e. A (f` x) e. B))
13	10, 12	syl5bb 530	. 2 |- (E.x e. A B = (/) -> (f e. X_x e. A B <-> f e. (/)))
14	13	eqrdv 1466	1 |- (E.x e. A B = (/) -> X_x e. A B = (/))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  E.wrex 1638  (/)c0 2270   Fn wfn 3167  ` cfv 3172  X_cixp 4331

This theorem is referenced by:  ac9s 4736

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-ixp 4332

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