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Theorem ac9s 9198
 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 8637). (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 9195 . . 3 (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
3 n0 3890 . . . 4 (X𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓X𝑥𝐴 𝐵)
4 vex 3176 . . . . . 6 𝑓 ∈ V
54elixp 7801 . . . . 5 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
65exbii 1764 . . . 4 (∃𝑓 𝑓X𝑥𝐴 𝐵 ↔ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
73, 6bitr2i 264 . . 3 (∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) ↔ X𝑥𝐴 𝐵 ≠ ∅)
82, 7sylib 207 . 2 (∀𝑥𝐴 𝐵 ≠ ∅ → X𝑥𝐴 𝐵 ≠ ∅)
9 ixpn0 7826 . 2 (X𝑥𝐴 𝐵 ≠ ∅ → ∀𝑥𝐴 𝐵 ≠ ∅)
108, 9impbii 198 1 (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874   Fn wfn 5799  ‘cfv 5804  Xcixp 7794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421  ax-ac2 9168 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-ixp 7795  df-en 7842  df-r1 8510  df-rank 8511  df-card 8648  df-ac 8822 This theorem is referenced by: (None)
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