Theorem ac9s 4764

Description: An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. This is a stronger version of the axiom in Enderton, with no existence requirement for the family of classes B(x) (achieved via the Collection Principle cp 4722).

Hypothesis

Ref	Expression
ac9.1	|- A e. V

Assertion

Ref	Expression
ac9s	|- (A.x e. A B =/= (/) <-> X_x e. A B =/= (/))

Distinct variable group:   x,A

Proof of Theorem ac9s

Step	Hyp	Ref	Expression
1		ac9.1	. . . 4 |- A e. V
2	1	ac6s4 4761	. . 3 |- (A.x e. A B =/= (/) -> E.f(f Fn A /\ A.x e. A (f` x) e. B))
3		ne0 2288	. . . 4 |- (X_x e. A B =/= (/) <-> E.f f e. X_x e. A B)
4		visset 1813	. . . . . 6 |- f e. V
5	4	elixp 4350	. . . . 5 |- (f e. X_x e. A B <-> (f Fn A /\ A.x e. A (f` x) e. B))
6	5	exbii 1051	. . . 4 |- (E.f f e. X_x e. A B <-> E.f(f Fn A /\ A.x e. A (f` x) e. B))
7	3, 6	bitr2 174	. . 3 |- (E.f(f Fn A /\ A.x e. A (f` x) e. B) <-> X_x e. A B =/= (/))
8	2, 7	sylib 198	. 2 |- (A.x e. A B =/= (/) -> X_x e. A B =/= (/))
9		ixp0 4361	. . . 4 |- (E.x e. A B = (/) -> X_x e. A B = (/))
10	9	con3i 98	. . 3 |- (-. X_x e. A B = (/) -> -. E.x e. A B = (/))
11		df-ne 1587	. . 3 |- (X_x e. A B =/= (/) <-> -. X_x e. A B = (/))
12		df-ne 1587	. . . . 5 |- (B =/= (/) <-> -. B = (/))
13	12	ralbii 1667	. . . 4 |- (A.x e. A B =/= (/) <-> A.x e. A -. B = (/))
14		ralnex 1653	. . . 4 |- (A.x e. A -. B = (/) <-> -. E.x e. A B = (/))
15	13, 14	bitr 173	. . 3 |- (A.x e. A B =/= (/) <-> -. E.x e. A B = (/))
16	10, 11, 15	3imtr4 219	. 2 |- (X_x e. A B =/= (/) -> A.x e. A B =/= (/))
17	8, 16	impbi 157	1 |- (A.x e. A B =/= (/) <-> X_x e. A B =/= (/))

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  A.wral 1645  E.wrex 1646  Vcvv 1811  (/)c0 2280   Fn wfn 3177  ` cfv 3182  X_cixp 4347

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-9 965  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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  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-f 3194  df-fv 3198  df-rdg 3932  df-ixp 4348  df-r1 4643  df-rank 4644

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