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Mirrors > Home > MPE Home > Th. List > ac9s | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ac9.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ac9s | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ac9.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | 1 | ac6s4 10289 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
3 | n0 4287 | . . . 4 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵) | |
4 | vex 3442 | . . . . . 6 ⊢ 𝑓 ∈ V | |
5 | 4 | elixp 8720 | . . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
6 | 5 | exbii 1848 | . . . 4 ⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) |
7 | 3, 6 | bitr2i 277 | . . 3 ⊢ (∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
8 | 2, 7 | sylib 217 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
9 | ixpn0 8746 | . 2 ⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) | |
10 | 8, 9 | impbii 208 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ X𝑥 ∈ 𝐴 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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