Theorem gchac 10595

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 3435	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2		omex 9555	. . . . . . . . 9 ⊢ ω ∈ V
3	1, 2	unex 7687	. . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V
4		ssun2 4108	. . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω)
5		ssdomg 8937	. . . . . . . 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 2846	. . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH)
9	3	pwex 5309	. . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V
10	9, 7	eleqtrrid 2846	. . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
11		gchacg 10594	. . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card)
12	6, 8, 10, 11	mp3an2i 1474	. . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
13	3	canth2 9058	. . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
14		sdomdom 8917	. . . . . . 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 9951	. . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
17	12, 15, 16	sylancl 592	. . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card)
18		ssun1 4107	. . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω)
19		ssnum 9952	. . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card)
20	17, 18, 19	sylancl 592	. . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card)
21	1	a1i 11	. . . 4 ⊢ (GCH = V → 𝑥 ∈ V)
22	20, 21	2thd 266	. . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
23	22	eqrdv 2737	. 2 ⊢ (GCH = V → dom card = V)
24		dfac10 10051	. 2 ⊢ (CHOICE ↔ dom card = V)
25	23, 24	sylibr 235	1 ⊢ (GCH = V → CHOICE)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∪ cun 3881   ⊆ wss 3883  𝒫 cpw 4529   class class class wbr 5072  dom cdm 5618  ωcom 7806   ≼ cdom 8881   ≺ csdm 8882  cardccrd 9850  CHOICEwac 10028  GCHcgch 10534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  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 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-oi 9415  df-har 9462  df-wdom 9470  df-cnf 9574  df-dju 9816  df-card 9854  df-ac 10029  df-fin4 10200  df-gch 10535

This theorem is referenced by: (None)