Theorem gch2 10590

Description: It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
gch2	⊢ (GCH = V ↔ ran ℵ ⊆ GCH)

Proof of Theorem gch2

Step	Hyp	Ref	Expression
1		ssv 3959	. . 3 ⊢ ran ℵ ⊆ V
2		sseq2 3961	. . 3 ⊢ (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
3	1, 2	mpbiri 258	. 2 ⊢ (GCH = V → ran ℵ ⊆ GCH)
4		cardidm 9875	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
5		iscard3 10007	. . . . . . . 8 ⊢ ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
6	4, 5	mpbi 230	. . . . . . 7 ⊢ (card‘𝑥) ∈ (ω ∪ ran ℵ)
7		elun 4106	. . . . . . 7 ⊢ ((card‘𝑥) ∈ (ω ∪ ran ℵ) ↔ ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ))
8	6, 7	mpbi 230	. . . . . 6 ⊢ ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ)
9		fingch 10538	. . . . . . . . 9 ⊢ Fin ⊆ GCH
10		nnfi 9096	. . . . . . . . 9 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
11	9, 10	sselid 3932	. . . . . . . 8 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
12	11	a1i 11	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13		ssel 3928	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ran ℵ → (card‘𝑥) ∈ GCH))
14	12, 13	jaod 860	. . . . . 6 ⊢ (ran ℵ ⊆ GCH → (((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ) → (card‘𝑥) ∈ GCH))
15	8, 14	mpi 20	. . . . 5 ⊢ (ran ℵ ⊆ GCH → (card‘𝑥) ∈ GCH)
16		vex 3445	. . . . . . 7 ⊢ 𝑥 ∈ V
17		alephon 9983	. . . . . . . . . . 11 ⊢ (ℵ‘suc 𝑥) ∈ On
18		simpr 484	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
19		simpl 482	. . . . . . . . . . . . 13 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → ran ℵ ⊆ GCH)
20		alephfnon 9979	. . . . . . . . . . . . . 14 ⊢ ℵ Fn On
21		fnfvelrn 7027	. . . . . . . . . . . . . 14 ⊢ ((ℵ Fn On ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
22	20, 18, 21	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
23	19, 22	sseldd 3935	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24		onsuc 7757	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
25	24	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26		fnfvelrn 7027	. . . . . . . . . . . . . 14 ⊢ ((ℵ Fn On ∧ suc 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
27	20, 25, 26	sylancr 588	. . . . . . . . . . . . 13 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
28	19, 27	sseldd 3935	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29		gchaleph2 10587	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
30	18, 23, 28, 29	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
31		isnumi 9862	. . . . . . . . . . 11 ⊢ (((ℵ‘suc 𝑥) ∈ On ∧ (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) → 𝒫 (ℵ‘𝑥) ∈ dom card)
32	17, 30, 31	sylancr 588	. . . . . . . . . 10 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝒫 (ℵ‘𝑥) ∈ dom card)
33	32	ralrimiva 3129	. . . . . . . . 9 ⊢ (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34		dfac12 10064	. . . . . . . . 9 ⊢ (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
35	33, 34	sylibr 234	. . . . . . . 8 ⊢ (ran ℵ ⊆ GCH → CHOICE)
36		dfac10 10052	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
37	35, 36	sylib 218	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → dom card = V)
38	16, 37	eleqtrrid 2844	. . . . . 6 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39		cardid2 9869	. . . . . 6 ⊢ (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40		engch 10543	. . . . . 6 ⊢ ((card‘𝑥) ≈ 𝑥 → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
41	38, 39, 40	3syl 18	. . . . 5 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
42	15, 41	mpbid 232	. . . 4 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ GCH)
43	16	a1i 11	. . . 4 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ V)
44	42, 43	2thd 265	. . 3 ⊢ (ran ℵ ⊆ GCH → (𝑥 ∈ GCH ↔ 𝑥 ∈ V))
45	44	eqrdv 2735	. 2 ⊢ (ran ℵ ⊆ GCH → GCH = V)
46	3, 45	impbii 209	1 ⊢ (GCH = V ↔ ran ℵ ⊆ GCH)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3441   ∪ cun 3900   ⊆ wss 3902  𝒫 cpw 4555   class class class wbr 5099  dom cdm 5625  ran crn 5626  Oncon0 6318  suc csuc 6320   Fn wfn 6488  ‘cfv 6493  ωcom 7810   ≈ cen 8884  Fincfn 8887  cardccrd 9851  ℵcale 9852  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-reg 9501  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-r1 9680  df-rank 9681  df-dju 9817  df-card 9855  df-aleph 9856  df-ac 10030  df-fin4 10201  df-gch 10536

This theorem is referenced by:  gch3  10591