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Theorem gchac 10618
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 3450 . . . . . . . . 9 𝑥 ∈ V
2 omex 9580 . . . . . . . . 9 ω ∈ V
31, 2unex 7681 . . . . . . . 8 (𝑥 ∪ ω) ∈ V
4 ssun2 4134 . . . . . . . 8 ω ⊆ (𝑥 ∪ ω)
5 ssdomg 8941 . . . . . . . 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 2845 . . . . . . 7 (GCH = V → (𝑥 ∪ ω) ∈ GCH)
93pwex 5336 . . . . . . . 8 𝒫 (𝑥 ∪ ω) ∈ V
109, 7eleqtrrid 2845 . . . . . . 7 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
11 gchacg 10617 . . . . . . 7 ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card)
126, 8, 10, 11mp3an2i 1467 . . . . . 6 (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
133canth2 9075 . . . . . . 7 (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
14 sdomdom 8921 . . . . . . 7 ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω))
1513, 14ax-mp 5 . . . . . 6 (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)
16 numdom 9975 . . . . . 6 ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
1712, 15, 16sylancl 587 . . . . 5 (GCH = V → (𝑥 ∪ ω) ∈ dom card)
18 ssun1 4133 . . . . 5 𝑥 ⊆ (𝑥 ∪ ω)
19 ssnum 9976 . . . . 5 (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card)
2017, 18, 19sylancl 587 . . . 4 (GCH = V → 𝑥 ∈ dom card)
211a1i 11 . . . 4 (GCH = V → 𝑥 ∈ V)
2220, 212thd 265 . . 3 (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
2322eqrdv 2735 . 2 (GCH = V → dom card = V)
24 dfac10 10074 . 2 (CHOICE ↔ dom card = V)
2523, 24sylibr 233 1 (GCH = V → CHOICE)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3446  cun 3909  wss 3911  𝒫 cpw 4561   class class class wbr 5106  dom cdm 5634  ωcom 7803  cdom 8882  csdm 8883  cardccrd 9872  CHOICEwac 10052  GCHcgch 10557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-seqom 8395  df-1o 8413  df-2o 8414  df-oadd 8417  df-omul 8418  df-oexp 8419  df-er 8649  df-map 8768  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-fsupp 9307  df-oi 9447  df-har 9494  df-wdom 9502  df-cnf 9599  df-dju 9838  df-card 9876  df-ac 10053  df-fin4 10224  df-gch 10558
This theorem is referenced by: (None)
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