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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 3499 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
2 | omex 9108 | . . . . . . . . 9 ⊢ ω ∈ V | |
3 | 1, 2 | unex 7471 | . . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V |
4 | ssun2 4151 | . . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω) | |
5 | ssdomg 8557 | . . . . . . . 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 2922 | . . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH) |
9 | 3 | pwex 5283 | . . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V |
10 | 9, 7 | eleqtrrid 2922 | . . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH) |
11 | gchacg 10104 | . . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card) | |
12 | 6, 8, 10, 11 | mp3an2i 1462 | . . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card) |
13 | 3 | canth2 8672 | . . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) |
14 | sdomdom 8539 | . . . . . . 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 9466 | . . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card) | |
17 | 12, 15, 16 | sylancl 588 | . . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card) |
18 | ssun1 4150 | . . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω) | |
19 | ssnum 9467 | . . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card) | |
20 | 17, 18, 19 | sylancl 588 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card) |
21 | 1 | a1i 11 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ V) |
22 | 20, 21 | 2thd 267 | . . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
23 | 22 | eqrdv 2821 | . 2 ⊢ (GCH = V → dom card = V) |
24 | dfac10 9565 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
25 | 23, 24 | sylibr 236 | 1 ⊢ (GCH = V → CHOICE) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 𝒫 cpw 4541 class class class wbr 5068 dom cdm 5557 ωcom 7582 ≼ cdom 8509 ≺ csdm 8510 cardccrd 9366 CHOICEwac 9543 GCHcgch 10044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-seqom 8086 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-oexp 8110 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-har 9024 df-wdom 9025 df-cnf 9127 df-dju 9332 df-card 9370 df-ac 9544 df-fin4 9711 df-gch 10045 |
This theorem is referenced by: (None) |
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