Theorem gch2 10666

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 3960	. . 3 ⊢ ran ℵ ⊆ V
2		sseq2 3962	. . 3 ⊢ (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
3	1, 2	mpbiri 261	. 2 ⊢ (GCH = V → ran ℵ ⊆ GCH)
4		cardidm 9952	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
5		iscard3 10084	. . . . . . . 8 ⊢ ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
6	4, 5	mpbi 233	. . . . . . 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 233	. . . . . 6 ⊢ ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ)
9		fingch 10614	. . . . . . . . 9 ⊢ Fin ⊆ GCH
10		nnfi 9150	. . . . . . . . 9 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
11	9, 10	sselid 3934	. . . . . . . 8 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
12	11	a1i 11	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13		ssel 3930	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ran ℵ → (card‘𝑥) ∈ GCH))
14	12, 13	jaod 872	. . . . . 6 ⊢ (ran ℵ ⊆ GCH → (((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ) → (card‘𝑥) ∈ GCH))
15	8, 14	mpi 21	. . . . 5 ⊢ (ran ℵ ⊆ GCH → (card‘𝑥) ∈ GCH)
16		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
17		alephon 10060	. . . . . . . . . . 11 ⊢ (ℵ‘suc 𝑥) ∈ On
18		simpr 489	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
19		simpl 487	. . . . . . . . . . . . 13 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → ran ℵ ⊆ GCH)
20		alephfnon 10056	. . . . . . . . . . . . . 14 ⊢ ℵ Fn On
21		fnfvelrn 7075	. . . . . . . . . . . . . 14 ⊢ ((ℵ Fn On ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
22	20, 18, 21	sylancr 598	. . . . . . . . . . . . 13 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
23	19, 22	sseldd 3937	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24		onsuc 7807	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
25	24	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26		fnfvelrn 7075	. . . . . . . . . . . . . 14 ⊢ ((ℵ Fn On ∧ suc 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
27	20, 25, 26	sylancr 598	. . . . . . . . . . . . 13 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
28	19, 27	sseldd 3937	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29		gchaleph2 10663	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
30	18, 23, 28, 29	syl3anc 1397	. . . . . . . . . . 11 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
31		isnumi 9939	. . . . . . . . . . 11 ⊢ (((ℵ‘suc 𝑥) ∈ On ∧ (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) → 𝒫 (ℵ‘𝑥) ∈ dom card)
32	17, 30, 31	sylancr 598	. . . . . . . . . 10 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝒫 (ℵ‘𝑥) ∈ dom card)
33	32	ralrimiva 3156	. . . . . . . . 9 ⊢ (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34		dfac12 10140	. . . . . . . . 9 ⊢ (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
35	33, 34	sylibr 237	. . . . . . . 8 ⊢ (ran ℵ ⊆ GCH → CHOICE)
36		dfac10 10128	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
37	35, 36	sylib 221	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → dom card = V)
38	16, 37	eleqtrrid 2869	. . . . . 6 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39		cardid2 9946	. . . . . 6 ⊢ (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40		engch 10619	. . . . . 6 ⊢ ((card‘𝑥) ≈ 𝑥 → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
41	38, 39, 40	3syl 19	. . . . 5 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
42	15, 41	mpbid 235	. . . 4 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ GCH)
43	16	a1i 11	. . . 4 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ V)
44	42, 43	2thd 268	. . 3 ⊢ (ran ℵ ⊆ GCH → (𝑥 ∈ GCH ↔ 𝑥 ∈ V))
45	44	eqrdv 2760	. 2 ⊢ (ran ℵ ⊆ GCH → GCH = V)
46	3, 45	impbii 212	1 ⊢ (GCH = V ↔ ran ℵ ⊆ GCH)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  𝒫 cpw 4561   class class class wbr 5108  dom cdm 5660  ran crn 5661  Oncon0 6360  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  ωcom 7860   ≈ cen 8938  Fincfn 8941  cardccrd 9928  ℵcale 9929  CHOICEwac 10106  GCHcgch 10611

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-reg 9552  ax-inf2 9608

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8155  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-seqom 8433  df-1o 8451  df-2o 8452  df-oadd 8455  df-omul 8456  df-oexp 8457  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9320  df-oi 9470  df-har 9517  df-wdom 9525  df-cnf 9629  df-r1 9734  df-rank 9735  df-dju 9894  df-card 9932  df-aleph 9933  df-ac 10107  df-fin4 10277  df-gch 10612

This theorem is used by:  gch3  10667