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Mirrors > Home > MPE Home > Th. List > acncc | Structured version Visualization version GIF version |
Description: An ax-cc 9655 equivalent: every set has choice sets of length ω. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
acncc | ⊢ AC ω = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3418 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | omex 8900 | . . . . 5 ⊢ ω ∈ V | |
3 | isacn 9264 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ ω ∈ V) → (𝑥 ∈ AC ω ↔ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 ω)∃𝑔∀𝑦 ∈ ω (𝑔‘𝑦) ∈ (𝑓‘𝑦))) | |
4 | 1, 2, 3 | mp2an 679 | . . . 4 ⊢ (𝑥 ∈ AC ω ↔ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 ω)∃𝑔∀𝑦 ∈ ω (𝑔‘𝑦) ∈ (𝑓‘𝑦)) |
5 | axcc2 9657 | . . . . 5 ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑦 ∈ ω ((𝑓‘𝑦) ≠ ∅ → (𝑔‘𝑦) ∈ (𝑓‘𝑦))) | |
6 | elmapi 8228 | . . . . . . . . . 10 ⊢ (𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 ω) → 𝑓:ω⟶(𝒫 𝑥 ∖ {∅})) | |
7 | ffvelrn 6674 | . . . . . . . . . . 11 ⊢ ((𝑓:ω⟶(𝒫 𝑥 ∖ {∅}) ∧ 𝑦 ∈ ω) → (𝑓‘𝑦) ∈ (𝒫 𝑥 ∖ {∅})) | |
8 | eldifsni 4596 | . . . . . . . . . . 11 ⊢ ((𝑓‘𝑦) ∈ (𝒫 𝑥 ∖ {∅}) → (𝑓‘𝑦) ≠ ∅) | |
9 | 7, 8 | syl 17 | . . . . . . . . . 10 ⊢ ((𝑓:ω⟶(𝒫 𝑥 ∖ {∅}) ∧ 𝑦 ∈ ω) → (𝑓‘𝑦) ≠ ∅) |
10 | 6, 9 | sylan 572 | . . . . . . . . 9 ⊢ ((𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 ω) ∧ 𝑦 ∈ ω) → (𝑓‘𝑦) ≠ ∅) |
11 | id 22 | . . . . . . . . 9 ⊢ (((𝑓‘𝑦) ≠ ∅ → (𝑔‘𝑦) ∈ (𝑓‘𝑦)) → ((𝑓‘𝑦) ≠ ∅ → (𝑔‘𝑦) ∈ (𝑓‘𝑦))) | |
12 | 10, 11 | syl5com 31 | . . . . . . . 8 ⊢ ((𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 ω) ∧ 𝑦 ∈ ω) → (((𝑓‘𝑦) ≠ ∅ → (𝑔‘𝑦) ∈ (𝑓‘𝑦)) → (𝑔‘𝑦) ∈ (𝑓‘𝑦))) |
13 | 12 | ralimdva 3127 | . . . . . . 7 ⊢ (𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 ω) → (∀𝑦 ∈ ω ((𝑓‘𝑦) ≠ ∅ → (𝑔‘𝑦) ∈ (𝑓‘𝑦)) → ∀𝑦 ∈ ω (𝑔‘𝑦) ∈ (𝑓‘𝑦))) |
14 | 13 | adantld 483 | . . . . . 6 ⊢ (𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 ω) → ((𝑔 Fn ω ∧ ∀𝑦 ∈ ω ((𝑓‘𝑦) ≠ ∅ → (𝑔‘𝑦) ∈ (𝑓‘𝑦))) → ∀𝑦 ∈ ω (𝑔‘𝑦) ∈ (𝑓‘𝑦))) |
15 | 14 | eximdv 1876 | . . . . 5 ⊢ (𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 ω) → (∃𝑔(𝑔 Fn ω ∧ ∀𝑦 ∈ ω ((𝑓‘𝑦) ≠ ∅ → (𝑔‘𝑦) ∈ (𝑓‘𝑦))) → ∃𝑔∀𝑦 ∈ ω (𝑔‘𝑦) ∈ (𝑓‘𝑦))) |
16 | 5, 15 | mpi 20 | . . . 4 ⊢ (𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑𝑚 ω) → ∃𝑔∀𝑦 ∈ ω (𝑔‘𝑦) ∈ (𝑓‘𝑦)) |
17 | 4, 16 | mprgbir 3103 | . . 3 ⊢ 𝑥 ∈ AC ω |
18 | 17, 1 | 2th 256 | . 2 ⊢ (𝑥 ∈ AC ω ↔ 𝑥 ∈ V) |
19 | 18 | eqriv 2775 | 1 ⊢ AC ω = V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 Vcvv 3415 ∖ cdif 3826 ∅c0 4178 𝒫 cpw 4422 {csn 4441 Fn wfn 6183 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ωcom 7396 ↑𝑚 cmap 8206 AC wacn 9161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cc 9655 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-er 8089 df-map 8208 df-en 8307 df-acn 9165 |
This theorem is referenced by: iunctb 9794 |
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