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 8776 | . . . . 5 ⊢ (𝐴 ≺ ω → 𝐴 ∈ Fin) | |
2 | 1 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → 𝐴 ∈ Fin) |
3 | simpr 487 | . . . 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 41483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≺ ω) → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) |
7 | simpr 487 | . . . . 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 1918 | . . 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 8540 | . . . 4 ⊢ (𝐴 ≈ ω ↔ (𝐴 ≼ ω ∧ ¬ 𝐴 ≺ ω)) | |
14 | 12, 13 | sylibr 236 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≺ ω) → 𝐴 ≈ ω) |
15 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ ω) → 𝐴 ≈ ω) | |
16 | 4 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≈ ω) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ ∅) |
17 | 15, 16 | axccd 41515 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≈ ω) → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥) |
18 | 14, 17 | syldan 593 | . 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 398 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∅c0 4291 class class class wbr 5066 Fn wfn 6350 ‘cfv 6355 ωcom 7580 ≈ cen 8506 ≼ cdom 8507 ≺ csdm 8508 Fincfn 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cc 9857 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 |
This theorem is referenced by: smflimlem6 43072 smfpimcc 43102 |
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