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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 3450 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
2 | omex 9580 | . . . . . . . . 9 ⊢ ω ∈ V | |
3 | 1, 2 | unex 7681 | . . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V |
4 | ssun2 4134 | . . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω) | |
5 | ssdomg 8941 | . . . . . . . 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 2845 | . . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH) |
9 | 3 | pwex 5336 | . . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V |
10 | 9, 7 | eleqtrrid 2845 | . . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH) |
11 | gchacg 10617 | . . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card) | |
12 | 6, 8, 10, 11 | mp3an2i 1467 | . . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card) |
13 | 3 | canth2 9075 | . . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) |
14 | sdomdom 8921 | . . . . . . 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 9975 | . . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card) | |
17 | 12, 15, 16 | sylancl 587 | . . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card) |
18 | ssun1 4133 | . . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω) | |
19 | ssnum 9976 | . . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card) | |
20 | 17, 18, 19 | sylancl 587 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card) |
21 | 1 | a1i 11 | . . . 4 ⊢ (GCH = V → 𝑥 ∈ V) |
22 | 20, 21 | 2thd 265 | . . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V)) |
23 | 22 | eqrdv 2735 | . 2 ⊢ (GCH = V → dom card = V) |
24 | dfac10 10074 | . 2 ⊢ (CHOICE ↔ dom card = V) | |
25 | 23, 24 | sylibr 233 | 1 ⊢ (GCH = V → CHOICE) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3446 ∪ cun 3909 ⊆ wss 3911 𝒫 cpw 4561 class class class wbr 5106 dom cdm 5634 ωcom 7803 ≼ cdom 8882 ≺ csdm 8883 cardccrd 9872 CHOICEwac 10052 GCHcgch 10557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-seqom 8395 df-1o 8413 df-2o 8414 df-oadd 8417 df-omul 8418 df-oexp 8419 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9307 df-oi 9447 df-har 9494 df-wdom 9502 df-cnf 9599 df-dju 9838 df-card 9876 df-ac 10053 df-fin4 10224 df-gch 10558 |
This theorem is referenced by: (None) |
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