Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3203 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
2 | omex 8540 | . . . . . . . . . 10 ⊢ ω ∈ V | |
3 | 1, 2 | unex 6956 | . . . . . . . . 9 ⊢ (𝑥 ∪ ω) ∈ V |
4 | ssun2 3777 | . . . . . . . . 9 ⊢ ω ⊆ (𝑥 ∪ ω) | |
5 | ssdomg 8001 | . . . . . . . . 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 2708 | . . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH) |
10 | 3 | pwex 4848 | . . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V |
11 | 10, 8 | syl5eleqr 2708 | . . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH) |
12 | gchacg 9502 | . . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card) | |
13 | 7, 9, 11, 12 | syl3anc 1326 | . . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card) |
14 | 3 | canth2 8113 | . . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) |
15 | sdomdom 7983 | . . . . . . 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 8861 | . . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card) | |
18 | 13, 16, 17 | sylancl 694 | . . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card) |
19 | ssun1 3776 | . . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω) | |
20 | ssnum 8862 | . . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card) | |
21 | 18, 19, 20 | sylancl 694 | . . . 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 2620 | . 2 ⊢ (GCH = V → dom card = V) |
25 | dfac10 8959 | . 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 1483 ∈ wcel 1990 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 𝒫 cpw 4158 class class class wbr 4653 dom cdm 5114 ωcom 7065 ≼ cdom 7953 ≺ csdm 7954 cardccrd 8761 CHOICEwac 8938 GCHcgch 9442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seqom 7543 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-oexp 7566 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-har 8463 df-wdom 8464 df-cnf 8559 df-card 8765 df-ac 8939 df-cda 8990 df-fin4 9109 df-gch 9443 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |