Theorem ac9s 4744

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 4702).

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 4741	. . 3 |- (A.x e. A B =/= (/) -> E.f(f Fn A /\ A.x e. A (f` x) e. B))
3		ne0 2284	. . . 4 |- (X_x e. A B =/= (/) <-> E.f f e. X_x e. A B)
4		visset 1809	. . . . . 6 |- f e. V
5	4	elixp 4340	. . . . 5 |- (f e. X_x e. A B <-> (f Fn A /\ A.x e. A (f` x) e. B))
6	5	exbii 1049	. . . 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 4351	. . . 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 1584	. . 3 |- (X_x e. A B =/= (/) <-> -. X_x e. A B = (/))
12		df-ne 1584	. . . . 5 |- (B =/= (/) <-> -. B = (/))
13	12	ralbii 1664	. . . 4 |- (A.x e. A B =/= (/) <-> A.x e. A -. B = (/))
14		ralnex 1650	. . . 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 954   e. wcel 956  E.wex 978   =/= wne 1582  A.wral 1642  E.wrex 1643  Vcvv 1807  (/)c0 2276   Fn wfn 3172  ` cfv 3177  X_cixp 4337

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-iin 2564  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-rdg 3923  df-ixp 4338  df-r1 4623  df-rank 4624

Copyright terms: Public domain