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Theorem gchac 10091
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 3495 . . . . . . . . 9 𝑥 ∈ V
2 omex 9094 . . . . . . . . 9 ω ∈ V
31, 2unex 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)))
63, 4, 5mp2 9 . . . . . . 7 ω ≼ (𝑥 ∪ ω)
7 id 22 . . . . . . . 8 (GCH = V → GCH = V)
83, 7eleqtrrid 2917 . . . . . . 7 (GCH = V → (𝑥 ∪ ω) ∈ GCH)
93pwex 5272 . . . . . . . 8 𝒫 (𝑥 ∪ ω) ∈ V
109, 7eleqtrrid 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)
126, 8, 10, 11mp3an2i 1457 . . . . . 6 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
133canth2 8658 . . . . . . 7 (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
14 sdomdom 8525 . . . . . . 7 ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω))
1513, 14ax-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)
1712, 15, 16sylancl 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)
2017, 18, 19sylancl 586 . . . 4 (GCH = V → 𝑥 ∈ dom card)
211a1i 11 . . . 4 (GCH = V → 𝑥 ∈ V)
2220, 212thd 266 . . 3 (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
2322eqrdv 2816 . 2 (GCH = V → dom card = V)
24 dfac10 9551 . 2 (CHOICE ↔ dom card = V)
2523, 24sylibr 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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