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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 3440 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
2 | omex 8957 | . . . . . . . . 9 ⊢ ω ∈ V | |
3 | 1, 2 | unex 7331 | . . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V |
4 | ssun2 4074 | . . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω) | |
5 | ssdomg 8408 | . . . . . . . 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 | syl5eleqr 2890 | . . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH) |
9 | 3 | pwex 5177 | . . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V |
10 | 9, 7 | syl5eleqr 2890 | . . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH) |
11 | gchacg 9953 | . . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card) | |
12 | 6, 8, 10, 11 | mp3an2i 1458 | . . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card) |
13 | 3 | canth2 8522 | . . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) |
14 | sdomdom 8390 | . . . . . . 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 9315 | . . . . . 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 4073 | . . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω) | |
19 | ssnum 9316 | . . . . 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 2793 | . 2 ⊢ (GCH = V → dom card = V) |
24 | dfac10 9414 | . 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 1522 ∈ wcel 2081 Vcvv 3437 ∪ cun 3861 ⊆ wss 3863 𝒫 cpw 4457 class class class wbr 4966 dom cdm 5448 ωcom 7441 ≼ cdom 8360 ≺ csdm 8361 cardccrd 9215 CHOICEwac 9392 GCHcgch 9893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-inf2 8955 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-se 5408 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-isom 6239 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-supp 7687 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-seqom 7940 df-1o 7958 df-2o 7959 df-oadd 7962 df-omul 7963 df-oexp 7964 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-fsupp 8685 df-oi 8825 df-har 8873 df-wdom 8874 df-cnf 8976 df-dju 9181 df-card 9219 df-ac 9393 df-fin4 9560 df-gch 9894 |
This theorem is referenced by: (None) |
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