Theorem gchac 9382

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 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
2		omex 8423	. . . . . . . . . 10 ⊢ ω ∈ V
3	1, 2	unex 6854	. . . . . . . . 9 ⊢ (𝑥 ∪ ω) ∈ V
4		ssun2 3739	. . . . . . . . 9 ⊢ ω ⊆ (𝑥 ∪ ω)
5		ssdomg 7887	. . . . . . . . 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 2695	. . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH)
10	3	pwex 4774	. . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V
11	10, 8	syl5eleqr 2695	. . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
12		gchacg 9381	. . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card)
13	7, 9, 11, 12	syl3anc 1318	. . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
14	3	canth2 7998	. . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
15		sdomdom 7869	. . . . . . 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 8744	. . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
18	13, 16, 17	sylancl 693	. . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card)
19		ssun1 3738	. . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω)
20		ssnum 8745	. . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card)
21	18, 19, 20	sylancl 693	. . . 4 ⊢ (GCH = V → 𝑥 ∈ dom card)
22	1	a1i 11	. . . 4 ⊢ (GCH = V → 𝑥 ∈ V)
23	21, 22	2thd 254	. . 3 ⊢ (GCH = V → (𝑥 ∈ dom card ↔ 𝑥 ∈ V))
24	23	eqrdv 2608	. 2 ⊢ (GCH = V → dom card = V)
25		dfac10 8842	. 2 ⊢ (CHOICE ↔ dom card = V)
26	24, 25	sylibr 223	1 ⊢ (GCH = V → CHOICE)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  dom cdm 5038  ωcom 6957   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644  CHOICEwac 8821  GCHcgch 9321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-har 8346  df-wdom 8347  df-cnf 8442  df-card 8648  df-ac 8822  df-cda 8873  df-fin4 8992  df-gch 9322

This theorem is referenced by: (None)