Theorem gchac 10590

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

Step	Hyp	Ref	Expression
1		vex 3442	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2		omex 9550	. . . . . . . . 9 ⊢ ω ∈ V
3	1, 2	unex 7687	. . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V
4		ssun2 4129	. . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω)
5		ssdomg 8935	. . . . . . . 8 ⊢ ((𝑥 ∪ ω) ∈ V → (ω ⊆ (𝑥 ∪ ω) → ω ≼ (𝑥 ∪ ω)))
6	3, 4, 5	mp2 9	. . . . . . 7 ⊢ ω ≼ (𝑥 ∪ ω)
7		id 22	. . . . . . . 8 ⊢ (GCH = V → GCH = V)
8	3, 7	eleqtrrid 2841	. . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH)
9	3	pwex 5323	. . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V
10	9, 7	eleqtrrid 2841	. . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
11		gchacg 10589	. . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card)
12	6, 8, 10, 11	mp3an2i 1468	. . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
13	3	canth2 9056	. . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
14		sdomdom 8915	. . . . . . 7 ⊢ ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω))
15	13, 14	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)
16		numdom 9946	. . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
17	12, 15, 16	sylancl 586	. . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card)
18		ssun1 4128	. . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω)
19		ssnum 9947	. . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card)
20	17, 18, 19	sylancl 586	. . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card)
21	1	a1i 11	. . . 4 ⊢ (GCH = V → 𝑥 ∈ V)
22	20, 21	2thd 265	. . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
23	22	eqrdv 2732	. 2 ⊢ (GCH = V → dom card = V)
24		dfac10 10046	. 2 ⊢ (CHOICE ↔ dom card = V)
25	23, 24	sylibr 234	1 ⊢ (GCH = V → CHOICE)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ∪ cun 3897   ⊆ wss 3899  𝒫 cpw 4552   class class class wbr 5096  dom cdm 5622  ωcom 7806   ≼ cdom 8879   ≺ csdm 8880  cardccrd 9845  CHOICEwac 10023  GCHcgch 10529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-seqom 8377  df-1o 8395  df-2o 8396  df-oadd 8399  df-omul 8400  df-oexp 8401  df-er 8633  df-map 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-fsupp 9263  df-oi 9413  df-har 9460  df-wdom 9468  df-cnf 9569  df-dju 9811  df-card 9849  df-ac 10024  df-fin4 10195  df-gch 10530

This theorem is referenced by: (None)