Theorem gchac 10604

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 3433	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2		omex 9564	. . . . . . . . 9 ⊢ ω ∈ V
3	1, 2	unex 7698	. . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V
4		ssun2 4119	. . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω)
5		ssdomg 8947	. . . . . . . 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 2843	. . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH)
9	3	pwex 5322	. . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V
10	9, 7	eleqtrrid 2843	. . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
11		gchacg 10603	. . . . . . 7 ⊢ ((ω ≼ (𝑥 ∪ ω) ∧ (𝑥 ∪ ω) ∈ GCH ∧ 𝒫 (𝑥 ∪ ω) ∈ GCH) → 𝒫 (𝑥 ∪ ω) ∈ dom card)
12	6, 8, 10, 11	mp3an2i 1469	. . . . . 6 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ dom card)
13	3	canth2 9068	. . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
14		sdomdom 8927	. . . . . . 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 9960	. . . . . 6 ⊢ ((𝒫 (𝑥 ∪ ω) ∈ dom card ∧ (𝑥 ∪ ω) ≼ 𝒫 (𝑥 ∪ ω)) → (𝑥 ∪ ω) ∈ dom card)
17	12, 15, 16	sylancl 587	. . . . 5 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ dom card)
18		ssun1 4118	. . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω)
19		ssnum 9961	. . . . 5 ⊢ (((𝑥 ∪ ω) ∈ dom card ∧ 𝑥 ⊆ (𝑥 ∪ ω)) → 𝑥 ∈ dom card)
20	17, 18, 19	sylancl 587	. . . 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 2734	. 2 ⊢ (GCH = V → dom card = V)
24		dfac10 10060	. 2 ⊢ (CHOICE ↔ dom card = V)
25	23, 24	sylibr 234	1 ⊢ (GCH = V → CHOICE)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889  𝒫 cpw 4541   class class class wbr 5085  dom cdm 5631  ωcom 7817   ≼ cdom 8891   ≺ csdm 8892  cardccrd 9859  CHOICEwac 10037  GCHcgch 10543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-seqom 8387  df-1o 8405  df-2o 8406  df-oadd 8409  df-omul 8410  df-oexp 8411  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-oi 9425  df-har 9472  df-wdom 9480  df-cnf 9583  df-dju 9825  df-card 9863  df-ac 10038  df-fin4 10209  df-gch 10544

This theorem is referenced by: (None)