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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 9284 | . . . . 5 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
2 | 1 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → 𝐴 ∈ Fin) |
3 | simpr 485 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≺ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
4 | axccd2.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) | |
5 | 4 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≺ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
6 | 2, 3, 5 | choicefi 43670 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) |
7 | simpr 485 | . . . . 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 1920 | . . 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 615 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) |
13 | bren2 8962 | . . . 4 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) | |
14 | 12, 13 | sylibr 233 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω) |
15 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ ω) → 𝐴 ≈ ω) | |
16 | 4 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≈ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
17 | 15, 16 | axccd 43699 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≈ ω) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
18 | 14, 17 | syldan 591 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
19 | 10, 18 | pm2.61dan 811 | 1 ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 ∅c0 4318 class class class wbr 5141 Fn wfn 6527 ‘cfv 6532 ωcom 7838 ≈ cen 8919 ≼ cdom 8920 ≺ csdm 8921 Fincfn 8922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cc 10412 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 |
This theorem is referenced by: smflimlem6 45265 smfpimcc 45297 |
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