Theorem gchac 10596

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 3445	. . . . . . . . 9 ⊢ 𝑥 ∈ V
2		omex 9556	. . . . . . . . 9 ⊢ ω ∈ V
3	1, 2	unex 7691	. . . . . . . 8 ⊢ (𝑥 ∪ ω) ∈ V
4		ssun2 4132	. . . . . . . 8 ⊢ ω ⊆ (𝑥 ∪ ω)
5		ssdomg 8941	. . . . . . . 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 2844	. . . . . . 7 ⊢ (GCH = V → (𝑥 ∪ ω) ∈ GCH)
9	3	pwex 5326	. . . . . . . 8 ⊢ 𝒫 (𝑥 ∪ ω) ∈ V
10	9, 7	eleqtrrid 2844	. . . . . . 7 ⊢ (GCH = V → 𝒫 (𝑥 ∪ ω) ∈ GCH)
11		gchacg 10595	. . . . . . 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 9062	. . . . . . 7 ⊢ (𝑥 ∪ ω) ≺ 𝒫 (𝑥 ∪ ω)
14		sdomdom 8921	. . . . . . 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 9952	. . . . . 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 4131	. . . . 5 ⊢ 𝑥 ⊆ (𝑥 ∪ ω)
19		ssnum 9953	. . . . 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 2735	. 2 ⊢ (GCH = V → dom card = V)
24		dfac10 10052	. 2 ⊢ (CHOICE ↔ dom card = V)
25	23, 24	sylibr 234	1 ⊢ (GCH = V → CHOICE)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∪ cun 3900   ⊆ wss 3902  𝒫 cpw 4555   class class class wbr 5099  dom cdm 5625  ωcom 7810   ≼ cdom 8885   ≺ csdm 8886  cardccrd 9851  CHOICEwac 10029  GCHcgch 10535

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 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-seqom 8381  df-1o 8399  df-2o 8400  df-oadd 8403  df-omul 8404  df-oexp 8405  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-oi 9419  df-har 9466  df-wdom 9474  df-cnf 9575  df-dju 9817  df-card 9855  df-ac 10030  df-fin4 10201  df-gch 10536

This theorem is referenced by: (None)