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

Step	Hyp	Ref	Expression
1		vex 3343	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
2		omex 8715	. . . . . . . . . 10 ⊢ ω ∈ V
3	1, 2	unex 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)))
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 2846	. . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH)
10	3	pwex 4997	. . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V
11	10, 8	syl5eleqr 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)
13	7, 9, 11, 12	syl3anc 1477	. . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
14	3	canth2 8280	. . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
15		sdomdom 8151	. . . . . . 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 9071	. . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
18	13, 16, 17	sylancl 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)
21	18, 19, 20	sylancl 697	. . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card)
22	1	a1i 11	. . . 4 ⊢ (GCH = V → 𝑥 ∈ V)
23	21, 22	2thd 255	. . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
24	23	eqrdv 2758	. 2 ⊢ (GCH = V → dom card = V)
25		dfac10 9171	. 2 ⊢ (CHOICE ↔ dom card = V)
26	24, 25	sylibr 224	1 ⊢ (GCH = V → CHOICE)

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)