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Theorem ac9s 10292
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 9690). (Contributed by NM, 29-Sep-2006.)
Hypothesis
Ref Expression
ac9.1 𝐴 ∈ V
Assertion
Ref Expression
ac9s (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ac9s
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ac9.1 . . . 4 𝐴 ∈ V
21ac6s4 10289 . . 3 (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
3 n0 4287 . . . 4 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
4 vex 3442 . . . . . 6 𝑓 ∈ V
54elixp 8720 . . . . 5 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
65exbii 1848 . . . 4 (∃𝑓 𝑓X𝑥𝐴 𝐵 ↔ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
73, 6bitr2i 277 . . 3 (∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ X𝑥𝐴 𝐵 ≠ ∅)
82, 7sylib 217 . 2 (∀𝑥𝐴 𝐵 ≠ ∅ → X𝑥𝐴 𝐵 ≠ ∅)
9 ixpn0 8746 . 2 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
108, 9impbii 208 1 (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wex 1779  wcel 2104  wne 2941  wral 3062  Vcvv 3438  c0 4263   Fn wfn 6450  cfv 6455  Xcixp 8713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5219  ax-sep 5233  ax-nul 5240  ax-pow 5298  ax-pr 5362  ax-un 7617  ax-reg 9392  ax-inf2 9440  ax-ac2 10262
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3278  df-reu 3279  df-rab 3280  df-v 3440  df-sbc 3723  df-csb 3839  df-dif 3896  df-un 3898  df-in 3900  df-ss 3910  df-pss 3912  df-nul 4264  df-if 4467  df-pw 4542  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4846  df-int 4888  df-iun 4934  df-iin 4935  df-br 5083  df-opab 5145  df-mpt 5166  df-tr 5200  df-id 5497  df-eprel 5503  df-po 5511  df-so 5512  df-fr 5552  df-se 5553  df-we 5554  df-xp 5603  df-rel 5604  df-cnv 5605  df-co 5606  df-dm 5607  df-rn 5608  df-res 5609  df-ima 5610  df-pred 6214  df-ord 6281  df-on 6282  df-lim 6283  df-suc 6284  df-iota 6407  df-fun 6457  df-fn 6458  df-f 6459  df-f1 6460  df-fo 6461  df-f1o 6462  df-fv 6463  df-isom 6464  df-riota 7261  df-ov 7307  df-om 7742  df-2nd 7861  df-frecs 8125  df-wrecs 8156  df-recs 8230  df-rdg 8269  df-ixp 8714  df-en 8762  df-r1 9563  df-rank 9564  df-card 9738  df-ac 9915
This theorem is referenced by: (None)
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