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

Step	Hyp	Ref	Expression
1		vex 3388	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
2		omex 8790	. . . . . . . . . 10 ⊢ ω ∈ V
3	1, 2	unex 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)))
6	3, 4, 5	mp2 9	. . . . . . . 8 ⊢ ω ≼ (𝑥 ∪ ω)
7	6	a1i 11	. . . . . . 7 ⊢ (GCH = V → ω ≼ (𝑥 ∪ ω))
8		id 22	. . . . . . . 8 ⊢ (GCH = V → GCH = V)
9	3, 8	syl5eleqr 2885	. . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH)
10	3	pwex 5050	. . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V
11	10, 8	syl5eleqr 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)
13	7, 9, 11, 12	syl3anc 1491	. . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
14	3	canth2 8355	. . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
15		sdomdom 8223	. . . . . . 7 ⊢ ((𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω) → (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω))
16	14, 15	ax-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)
18	13, 16, 17	sylancl 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)
21	18, 19, 20	sylancl 581	. . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card)
22	1	a1i 11	. . . 4 ⊢ (GCH = V → 𝑥 ∈ V)
23	21, 22	2thd 257	. . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
24	23	eqrdv 2797	. 2 ⊢ (GCH = V → dom card = V)
25		dfac10 9247	. 2 ⊢ (CHOICE ↔ dom card = V)
26	24, 25	sylibr 226	1 ⊢ (GCH = V → CHOICE)

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)