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Theorem gchac 10096
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.)
Assertion
Ref Expression
gchac (GCH = V → CHOICE)

Proof of Theorem gchac
StepHypRef Expression
1 vex 3447 . . . . . . . . 9 𝑥 ∈ V
2 omex 9094 . . . . . . . . 9 ω ∈ V
31, 2unex 7453 . . . . . . . 8 (𝑥 ∪ ω) ∈ V
4 ssun2 4103 . . . . . . . 8 ω ⊆ (𝑥 ∪ ω)
5 ssdomg 8542 . . . . . . . 8 ((𝑥 ∪ ω) ∈ V → (ω ⊆ (𝑥 ∪ ω) → ω ≼ (𝑥 ∪ ω)))
63, 4, 5mp2 9 . . . . . . 7 ω ≼ (𝑥 ∪ ω)
7 id 22 . . . . . . . 8 (GCH = V → GCH = V)
83, 7eleqtrrid 2900 . . . . . . 7 (GCH = V → (𝑥 ∪ ω) ∈ GCH)
93pwex 5249 . . . . . . . 8 𝒫 (𝑥 ∪ ω) ∈ V
109, 7eleqtrrid 2900 . . . . . . 7 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
11 gchacg 10095 . . . . . . 7 ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card)
126, 8, 10, 11mp3an2i 1463 . . . . . 6 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
133canth2 8658 . . . . . . 7 (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
14 sdomdom 8524 . . . . . . 7 ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω))
1513, 14ax-mp 5 . . . . . 6 (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)
16 numdom 9453 . . . . . 6 ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
1712, 15, 16sylancl 589 . . . . 5 (GCH = V → (𝑥 ∪ ω) ∈ dom card)
18 ssun1 4102 . . . . 5 𝑥 ⊆ (𝑥 ∪ ω)
19 ssnum 9454 . . . . 5 (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card)
2017, 18, 19sylancl 589 . . . 4 (GCH = V → 𝑥 ∈ dom card)
211a1i 11 . . . 4 (GCH = V → 𝑥 ∈ V)
2220, 212thd 268 . . 3 (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
2322eqrdv 2799 . 2 (GCH = V → dom card = V)
24 dfac10 9552 . 2 (CHOICE ↔ dom card = V)
2523, 24sylibr 237 1 (GCH = V → CHOICE)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  Vcvv 3444  cun 3882  wss 3884  𝒫 cpw 4500   class class class wbr 5033  dom cdm 5523  ωcom 7564  cdom 8494  csdm 8495  cardccrd 9352  CHOICEwac 9530  GCHcgch 10035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-oexp 8095  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-har 9009  df-wdom 9017  df-cnf 9113  df-dju 9318  df-card 9356  df-ac 9531  df-fin4 9702  df-gch 10036
This theorem is referenced by: (None)
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