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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > axccd2 | Structured version Visualization version GIF version |
Description: An alternative version of the axiom of countable choice. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
axccd2.1 | ⊢ (𝜑 → 𝐴 ≼ ω) |
axccd2.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
Ref | Expression |
---|---|
axccd2 | ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfinite2 9301 | . . . . 5 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
2 | 1 | adantl 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → 𝐴 ∈ Fin) |
3 | simpr 486 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≺ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
4 | axccd2.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) | |
5 | 4 | adantlr 714 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≺ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
6 | 2, 3, 5 | choicefi 43899 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) |
7 | simpr 486 | . . . . 5 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) | |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) |
9 | 8 | eximdv 1921 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → (∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) |
10 | 6, 9 | mpd 15 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
11 | axccd2.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ≼ ω) | |
12 | 11 | anim1i 616 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) |
13 | bren2 8979 | . . . 4 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) | |
14 | 12, 13 | sylibr 233 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω) |
15 | simpr 486 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ ω) → 𝐴 ≈ ω) | |
16 | 4 | adantlr 714 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≈ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
17 | 15, 16 | axccd 43928 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≈ ω) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
18 | 14, 17 | syldan 592 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
19 | 10, 18 | pm2.61dan 812 | 1 ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∅c0 4323 class class class wbr 5149 Fn wfn 6539 ‘cfv 6544 ωcom 7855 ≈ cen 8936 ≼ cdom 8937 ≺ csdm 8938 Fincfn 8939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cc 10430 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 |
This theorem is referenced by: smflimlem6 45492 smfpimcc 45524 |
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