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 8103 | . . . . 5 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
2 | 1 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → 𝐴 ∈ Fin) |
3 | simpr 476 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≺ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
4 | axccd2.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) | |
5 | 4 | adantlr 747 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≺ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
6 | 2, 3, 5 | choicefi 38387 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) |
7 | simpr 476 | . . . . 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 1833 | . . 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 590 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) |
13 | bren2 7872 | . . . 4 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) | |
14 | 12, 13 | sylibr 223 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω) |
15 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ ω) → 𝐴 ≈ ω) | |
16 | 4 | adantlr 747 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≈ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
17 | 15, 16 | axccd 38424 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≈ ω) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
18 | 14, 17 | syldan 486 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
19 | 10, 18 | pm2.61dan 828 | 1 ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∅c0 3874 class class class wbr 4583 Fn wfn 5799 ‘cfv 5804 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 Fincfn 7841 |
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-cc 9140 |
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-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-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-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-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 |
This theorem is referenced by: smflimlem6 39662 |
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