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Theorem gchac 9715
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 3343 . . . . . . . . . 10 𝑥 ∈ V
2 omex 8715 . . . . . . . . . 10 ω ∈ V
31, 2unex 7122 . . . . . . . . 9 (𝑥 ∪ ω) ∈ V
4 ssun2 3920 . . . . . . . . 9 ω ⊆ (𝑥 ∪ ω)
5 ssdomg 8169 . . . . . . . . 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 2846 . . . . . . 7 (GCH = V → (𝑥 ∪ ω) ∈ GCH)
103pwex 4997 . . . . . . . 8 𝒫 (𝑥 ∪ ω) ∈ V
1110, 8syl5eleqr 2846 . . . . . . 7 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
12 gchacg 9714 . . . . . . 7 ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card)
137, 9, 11, 12syl3anc 1477 . . . . . 6 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
143canth2 8280 . . . . . . 7 (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
15 sdomdom 8151 . . . . . . 7 ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω))
1614, 15ax-mp 5 . . . . . 6 (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)
17 numdom 9071 . . . . . 6 ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
1813, 16, 17sylancl 697 . . . . 5 (GCH = V → (𝑥 ∪ ω) ∈ dom card)
19 ssun1 3919 . . . . 5 𝑥 ⊆ (𝑥 ∪ ω)
20 ssnum 9072 . . . . 5 (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card)
2118, 19, 20sylancl 697 . . . 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 2758 . 2 (GCH = V → dom card = V)
25 dfac10 9171 . 2 (CHOICE ↔ dom card = V)
2624, 25sylibr 224 1 (GCH = V → CHOICE)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  wss 3715  𝒫 cpw 4302   class class class wbr 4804  dom cdm 5266  ωcom 7231  cdom 8121  csdm 8122  cardccrd 8971  CHOICEwac 9148  GCHcgch 9654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-supp 7465  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-seqom 7713  df-1o 7730  df-2o 7731  df-oadd 7734  df-omul 7735  df-oexp 7736  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fsupp 8443  df-oi 8582  df-har 8630  df-wdom 8631  df-cnf 8734  df-card 8975  df-ac 9149  df-cda 9202  df-fin4 9321  df-gch 9655
This theorem is referenced by: (None)
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