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Theorem gchac 9791
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 3388 . . . . . . . . . 10 𝑥 ∈ V
2 omex 8790 . . . . . . . . . 10 ω ∈ V
31, 2unex 7190 . . . . . . . . 9 (𝑥 ∪ ω) ∈ V
4 ssun2 3975 . . . . . . . . 9 ω ⊆ (𝑥 ∪ ω)
5 ssdomg 8241 . . . . . . . . 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 2885 . . . . . . 7 (GCH = V → (𝑥 ∪ ω) ∈ GCH)
103pwex 5050 . . . . . . . 8 𝒫 (𝑥 ∪ ω) ∈ V
1110, 8syl5eleqr 2885 . . . . . . 7 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
12 gchacg 9790 . . . . . . 7 ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card)
137, 9, 11, 12syl3anc 1491 . . . . . 6 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
143canth2 8355 . . . . . . 7 (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
15 sdomdom 8223 . . . . . . 7 ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω))
1614, 15ax-mp 5 . . . . . 6 (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)
17 numdom 9147 . . . . . 6 ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
1813, 16, 17sylancl 581 . . . . 5 (GCH = V → (𝑥 ∪ ω) ∈ dom card)
19 ssun1 3974 . . . . 5 𝑥 ⊆ (𝑥 ∪ ω)
20 ssnum 9148 . . . . 5 (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card)
2118, 19, 20sylancl 581 . . . 4 (GCH = V → 𝑥 ∈ dom card)
221a1i 11 . . . 4 (GCH = V → 𝑥 ∈ V)
2321, 222thd 257 . . 3 (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
2423eqrdv 2797 . 2 (GCH = V → dom card = V)
25 dfac10 9247 . 2 (CHOICE ↔ dom card = V)
2624, 25sylibr 226 1 (GCH = V → CHOICE)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  Vcvv 3385  cun 3767  wss 3769  𝒫 cpw 4349   class class class wbr 4843  dom cdm 5312  ωcom 7299  cdom 8193  csdm 8194  cardccrd 9047  CHOICEwac 9224  GCHcgch 9730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-inf2 8788
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-supp 7533  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-seqom 7782  df-1o 7799  df-2o 7800  df-oadd 7803  df-omul 7804  df-oexp 7805  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fsupp 8518  df-oi 8657  df-har 8705  df-wdom 8706  df-cnf 8809  df-card 9051  df-ac 9225  df-cda 9278  df-fin4 9397  df-gch 9731
This theorem is referenced by: (None)
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