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Theorem gchac 9447
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 3189 . . . . . . . . . 10 𝑥 ∈ V
2 omex 8484 . . . . . . . . . 10 ω ∈ V
31, 2unex 6909 . . . . . . . . 9 (𝑥 ∪ ω) ∈ V
4 ssun2 3755 . . . . . . . . 9 ω ⊆ (𝑥 ∪ ω)
5 ssdomg 7945 . . . . . . . . 9 ((𝑥 ∪ ω) ∈ V → (ω ⊆ (𝑥 ∪ ω) → ω ≼ (𝑥 ∪ ω)))
63, 4, 5mp2 9 . . . . . . . 8 ω ≼ (𝑥 ∪ ω)
76a1i 11 . . . . . . 7 (GCH = V → ω ≼ (𝑥 ∪ ω))
8 id 22 . . . . . . . 8 (GCH = V → GCH = V)
93, 8syl5eleqr 2705 . . . . . . 7 (GCH = V → (𝑥 ∪ ω) ∈ GCH)
103pwex 4808 . . . . . . . 8 𝒫 (𝑥 ∪ ω) ∈ V
1110, 8syl5eleqr 2705 . . . . . . 7 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
12 gchacg 9446 . . . . . . 7 ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card)
137, 9, 11, 12syl3anc 1323 . . . . . 6 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
143canth2 8057 . . . . . . 7 (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
15 sdomdom 7927 . . . . . . 7 ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω))
1614, 15ax-mp 5 . . . . . 6 (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)
17 numdom 8805 . . . . . 6 ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
1813, 16, 17sylancl 693 . . . . 5 (GCH = V → (𝑥 ∪ ω) ∈ dom card)
19 ssun1 3754 . . . . 5 𝑥 ⊆ (𝑥 ∪ ω)
20 ssnum 8806 . . . . 5 (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card)
2118, 19, 20sylancl 693 . . . 4 (GCH = V → 𝑥 ∈ dom card)
221a1i 11 . . . 4 (GCH = V → 𝑥 ∈ V)
2321, 222thd 255 . . 3 (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
2423eqrdv 2619 . 2 (GCH = V → dom card = V)
25 dfac10 8903 . 2 (CHOICE ↔ dom card = V)
2624, 25sylibr 224 1 (GCH = V → CHOICE)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  cun 3553  wss 3555  𝒫 cpw 4130   class class class wbr 4613  dom cdm 5074  ωcom 7012  cdom 7897  csdm 7898  cardccrd 8705  CHOICEwac 8882  GCHcgch 9386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-har 8407  df-wdom 8408  df-cnf 8503  df-card 8709  df-ac 8883  df-cda 8934  df-fin4 9053  df-gch 9387
This theorem is referenced by: (None)
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