Theorem gchac 10695

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 3463	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2		omex 9657	. . . . . . . . 9 ⊢ ω ∈ V
3	1, 2	unex 7738	. . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V
4		ssun2 4154	. . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω)
5		ssdomg 9014	. . . . . . . 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 5350	. . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V
10	9, 7	eleqtrrid 2841	. . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
11		gchacg 10694	. . . . . . 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 9144	. . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
14		sdomdom 8994	. . . . . . 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 10052	. . . . . 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 4153	. . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω)
19		ssnum 10053	. . . . 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 2733	. 2 ⊢ (GCH = V → dom card = V)
24		dfac10 10152	. 2 ⊢ (CHOICE ↔ dom card = V)
25	23, 24	sylibr 234	1 ⊢ (GCH = V → CHOICE)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924   ⊆ wss 3926  𝒫 cpw 4575   class class class wbr 5119  dom cdm 5654  ωcom 7861   ≼ cdom 8957   ≺ csdm 8958  cardccrd 9949  CHOICEwac 10129  GCHcgch 10634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-seqom 8462  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-oexp 8486  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-oi 9524  df-har 9571  df-wdom 9579  df-cnf 9676  df-dju 9915  df-card 9953  df-ac 10130  df-fin4 10301  df-gch 10635

This theorem is referenced by: (None)