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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 3495 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
2 | omex 9094 | . . . . . . . . 9 ⊢ ω ∈ V | |
3 | 1, 2 | unex 7458 | . . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V |
4 | ssun2 4146 | . . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω) | |
5 | ssdomg 8543 | . . . . . . . 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 2917 | . . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH) |
9 | 3 | pwex 5272 | . . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V |
10 | 9, 7 | eleqtrrid 2917 | . . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH) |
11 | gchacg 10090 | . . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card) | |
12 | 6, 8, 10, 11 | mp3an2i 1457 | . . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card) |
13 | 3 | canth2 8658 | . . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) |
14 | sdomdom 8525 | . . . . . . 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 9452 | . . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card) | |
17 | 12, 15, 16 | sylancl 586 | . . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card) |
18 | ssun1 4145 | . . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω) | |
19 | ssnum 9453 | . . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card) | |
20 | 17, 18, 19 | sylancl 586 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card) |
21 | 1 | a1i 11 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ V) |
22 | 20, 21 | 2thd 266 | . . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
23 | 22 | eqrdv 2816 | . 2 ⊢ (GCH = V → dom card = V) |
24 | dfac10 9551 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
25 | 23, 24 | sylibr 235 | 1 ⊢ (GCH = V → CHOICE) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 𝒫 cpw 4535 class class class wbr 5057 dom cdm 5548 ωcom 7569 ≼ cdom 8495 ≺ csdm 8496 cardccrd 9352 CHOICEwac 9529 GCHcgch 10030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-seqom 8073 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-oexp 8097 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-har 9010 df-wdom 9011 df-cnf 9113 df-dju 9318 df-card 9356 df-ac 9530 df-fin4 9697 df-gch 10031 |
This theorem is referenced by: (None) |
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