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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 3388 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
2 | omex 8790 | . . . . . . . . . 10 ⊢ ω ∈ V | |
3 | 1, 2 | unex 7190 | . . . . . . . . 9 ⊢ (𝑥 ∪ ω) ∈ V |
4 | ssun2 3975 | . . . . . . . . 9 ⊢ ω ⊆ (𝑥 ∪ ω) | |
5 | ssdomg 8241 | . . . . . . . . 9 ⊢ ((𝑥 ∪ ω) ∈ V → (ω ⊆ (𝑥 ∪ ω) → ω ≼ (𝑥 ∪ ω))) | |
6 | 3, 4, 5 | mp2 9 | . . . . . . . 8 ⊢ ω ≼ (𝑥 ∪ ω) |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (GCH = V → ω ≼ (𝑥 ∪ ω)) |
8 | id 22 | . . . . . . . 8 ⊢ (GCH = V → GCH = V) | |
9 | 3, 8 | syl5eleqr 2885 | . . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH) |
10 | 3 | pwex 5050 | . . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V |
11 | 10, 8 | syl5eleqr 2885 | . . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH) |
12 | gchacg 9790 | . . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card) | |
13 | 7, 9, 11, 12 | syl3anc 1491 | . . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card) |
14 | 3 | canth2 8355 | . . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) |
15 | sdomdom 8223 | . . . . . . 7 ⊢ ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 ⊢ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω) |
17 | numdom 9147 | . . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card) | |
18 | 13, 16, 17 | sylancl 581 | . . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card) |
19 | ssun1 3974 | . . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω) | |
20 | ssnum 9148 | . . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card) | |
21 | 18, 19, 20 | sylancl 581 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card) |
22 | 1 | a1i 11 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ V) |
23 | 21, 22 | 2thd 257 | . . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
24 | 23 | eqrdv 2797 | . 2 ⊢ (GCH = V → dom card = V) |
25 | dfac10 9247 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
26 | 24, 25 | sylibr 226 | 1 ⊢ (GCH = V → CHOICE) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 Vcvv 3385 ∪ cun 3767 ⊆ wss 3769 𝒫 cpw 4349 class class class wbr 4843 dom cdm 5312 ωcom 7299 ≼ cdom 8193 ≺ csdm 8194 cardccrd 9047 CHOICEwac 9224 GCHcgch 9730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-seqom 7782 df-1o 7799 df-2o 7800 df-oadd 7803 df-omul 7804 df-oexp 7805 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-oi 8657 df-har 8705 df-wdom 8706 df-cnf 8809 df-card 9051 df-ac 9225 df-cda 9278 df-fin4 9397 df-gch 9731 |
This theorem is referenced by: (None) |
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