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Mirrors > Home > MPE Home > Th. List > acncc | Structured version Visualization version GIF version |
Description: An ax-cc 10014 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 3402 | . . . . 5 ⊢ 𝑥 ∈ V | |
2 | omex 9236 | . . . . 5 ⊢ ω ∈ V | |
3 | isacn 9623 | . . . . 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 692 | . . . 4 ⊢ (𝑥 ∈ AC ω ↔ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m ω)∃𝑔∀𝑦 ∈ ω (𝑔‘𝑦) ∈ (𝑓‘𝑦)) |
5 | axcc2 10016 | . . . . 5 ⊢ ∃𝑔(𝑔 Fn ω ∧ ∀𝑦 ∈ ω ((𝑓‘𝑦) ≠ ∅ → (𝑔‘𝑦) ∈ (𝑓‘𝑦))) | |
6 | elmapi 8508 | . . . . . . . . . 10 ⊢ (𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m ω) → 𝑓:ω⟶(𝒫 𝑥 ∖ {∅})) | |
7 | ffvelrn 6880 | . . . . . . . . . . 11 ⊢ ((𝑓:ω⟶(𝒫 𝑥 ∖ {∅}) ∧ 𝑦 ∈ ω) → (𝑓‘𝑦) ∈ (𝒫 𝑥 ∖ {∅})) | |
8 | eldifsni 4689 | . . . . . . . . . . 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 583 | . . . . . . . . 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 3090 | . . . . . . 7 ⊢ (𝑓 ∈ ((𝒫 𝑥 ∖ {∅}) ↑m ω) → (∀𝑦 ∈ ω ((𝑓‘𝑦) ≠ ∅ → (𝑔‘𝑦) ∈ (𝑓‘𝑦)) → ∀𝑦 ∈ ω (𝑔‘𝑦) ∈ (𝑓‘𝑦))) |
14 | 13 | adantld 494 | . . . . . 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 1925 | . . . . 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 3066 | . . 3 ⊢ 𝑥 ∈ AC ω |
18 | 17, 1 | 2th 267 | . 2 ⊢ (𝑥 ∈ AC ω ↔ 𝑥 ∈ V) |
19 | 18 | eqriv 2733 | 1 ⊢ AC ω = V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 Vcvv 3398 ∖ cdif 3850 ∅c0 4223 𝒫 cpw 4499 {csn 4527 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ωcom 7622 ↑m cmap 8486 AC wacn 9519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cc 10014 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-er 8369 df-map 8488 df-en 8605 df-acn 9523 |
This theorem is referenced by: iunctb 10153 |
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