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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 3343 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
2 | omex 8715 | . . . . . . . . . 10 ⊢ ω ∈ V | |
3 | 1, 2 | unex 7122 | . . . . . . . . 9 ⊢ (𝑥 ∪ ω) ∈ V |
4 | ssun2 3920 | . . . . . . . . 9 ⊢ ω ⊆ (𝑥 ∪ ω) | |
5 | ssdomg 8169 | . . . . . . . . 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 2846 | . . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH) |
10 | 3 | pwex 4997 | . . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V |
11 | 10, 8 | syl5eleqr 2846 | . . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH) |
12 | gchacg 9714 | . . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card) | |
13 | 7, 9, 11, 12 | syl3anc 1477 | . . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card) |
14 | 3 | canth2 8280 | . . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) |
15 | sdomdom 8151 | . . . . . . 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 9071 | . . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card) | |
18 | 13, 16, 17 | sylancl 697 | . . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card) |
19 | ssun1 3919 | . . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω) | |
20 | ssnum 9072 | . . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card) | |
21 | 18, 19, 20 | sylancl 697 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card) |
22 | 1 | a1i 11 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ V) |
23 | 21, 22 | 2thd 255 | . . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
24 | 23 | eqrdv 2758 | . 2 ⊢ (GCH = V → dom card = V) |
25 | dfac10 9171 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
26 | 24, 25 | sylibr 224 | 1 ⊢ (GCH = V → CHOICE) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∪ cun 3713 ⊆ wss 3715 𝒫 cpw 4302 class class class wbr 4804 dom cdm 5266 ωcom 7231 ≼ cdom 8121 ≺ csdm 8122 cardccrd 8971 CHOICEwac 9148 GCHcgch 9654 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-seqom 7713 df-1o 7730 df-2o 7731 df-oadd 7734 df-omul 7735 df-oexp 7736 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-oi 8582 df-har 8630 df-wdom 8631 df-cnf 8734 df-card 8975 df-ac 9149 df-cda 9202 df-fin4 9321 df-gch 9655 |
This theorem is referenced by: (None) |
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