| 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 9334 | . . . . 5 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
| 2 | 1 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → 𝐴 ∈ Fin) |
| 3 | simpr 484 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≺ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
| 4 | axccd2.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) | |
| 5 | 4 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≺ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
| 6 | 2, 3, 5 | choicefi 45205 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) |
| 7 | simpr 484 | . . . . 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 1917 | . . 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 9023 | . . . 4 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) | |
| 14 | 12, 13 | sylibr 234 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω) |
| 15 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ ω) → 𝐴 ≈ ω) | |
| 16 | 4 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≈ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
| 17 | 15, 16 | axccd 45234 | . . 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 813 | 1 ⊢ (𝜑 → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∅c0 4333 class class class wbr 5143 Fn wfn 6556 ‘cfv 6561 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 Fincfn 8985 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cc 10475 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 |
| This theorem is referenced by: smflimlem6 46791 smfpimcc 46823 |
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