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Mirrors > Home > MPE Home > Th. List > gchac | Structured version Visualization version GIF version |
Description: The Generalized Continuum Hypothesis implies the Axiom of Choice. The original proof is due to Sierpiński (1947); we use a refinement of Sierpiński's result due to Specker. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
gchac | ⊢ (GCH = V → CHOICE) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3402 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
2 | omex 9236 | . . . . . . . . 9 ⊢ ω ∈ V | |
3 | 1, 2 | unex 7509 | . . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V |
4 | ssun2 4073 | . . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω) | |
5 | ssdomg 8652 | . . . . . . . 8 ⊢ ((𝑥 ∪ ω) ∈ V → (ω ⊆ (𝑥 ∪ ω) → ω ≼ (𝑥 ∪ ω))) | |
6 | 3, 4, 5 | mp2 9 | . . . . . . 7 ⊢ ω ≼ (𝑥 ∪ ω) |
7 | id 22 | . . . . . . . 8 ⊢ (GCH = V → GCH = V) | |
8 | 3, 7 | eleqtrrid 2838 | . . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH) |
9 | 3 | pwex 5258 | . . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V |
10 | 9, 7 | eleqtrrid 2838 | . . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH) |
11 | gchacg 10259 | . . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card) | |
12 | 6, 8, 10, 11 | mp3an2i 1468 | . . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card) |
13 | 3 | canth2 8777 | . . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) |
14 | sdomdom 8634 | . . . . . . 7 ⊢ ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 ⊢ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω) |
16 | numdom 9617 | . . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card) | |
17 | 12, 15, 16 | sylancl 589 | . . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card) |
18 | ssun1 4072 | . . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω) | |
19 | ssnum 9618 | . . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card) | |
20 | 17, 18, 19 | sylancl 589 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card) |
21 | 1 | a1i 11 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ V) |
22 | 20, 21 | 2thd 268 | . . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
23 | 22 | eqrdv 2734 | . 2 ⊢ (GCH = V → dom card = V) |
24 | dfac10 9716 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
25 | 23, 24 | sylibr 237 | 1 ⊢ (GCH = V → CHOICE) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∪ cun 3851 ⊆ wss 3853 𝒫 cpw 4499 class class class wbr 5039 dom cdm 5536 ωcom 7622 ≼ cdom 8602 ≺ csdm 8603 cardccrd 9516 CHOICEwac 9694 GCHcgch 10199 |
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 |
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-rmo 3059 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-int 4846 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-se 5495 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-pred 6140 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-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-seqom 8162 df-1o 8180 df-2o 8181 df-oadd 8184 df-omul 8185 df-oexp 8186 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-oi 9104 df-har 9151 df-wdom 9159 df-cnf 9255 df-dju 9482 df-card 9520 df-ac 9695 df-fin4 9866 df-gch 10200 |
This theorem is referenced by: (None) |
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