Theorem gch2 10587

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 3941	. . 3 ⊢ ran ℵ ⊆ V
2		sseq2 3943	. . 3 ⊢ (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
3	1, 2	mpbiri 258	. 2 ⊢ (GCH = V → ran ℵ ⊆ GCH)
4		cardidm 9872	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
5		iscard3 10004	. . . . . . . 8 ⊢ ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
6	4, 5	mpbi 230	. . . . . . 7 ⊢ (card‘𝑥) ∈ (ω ∪ ran ℵ)
7		elun 4085	. . . . . . 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 10535	. . . . . . . . 9 ⊢ Fin ⊆ GCH
10		nnfi 9091	. . . . . . . . 9 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
11	9, 10	sselid 3915	. . . . . . . 8 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
12	11	a1i 11	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13		ssel 3911	. . . . . . 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 3431	. . . . . . 7 ⊢ 𝑥 ∈ V
17		alephon 9980	. . . . . . . . . . 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 9976	. . . . . . . . . . . . . 14 ⊢ ℵ Fn On
21		fnfvelrn 7021	. . . . . . . . . . . . . 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 3918	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24		onsuc 7753	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
25	24	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26		fnfvelrn 7021	. . . . . . . . . . . . . 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 3918	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29		gchaleph2 10584	. . . . . . . . . . . 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 9859	. . . . . . . . . . 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 3127	. . . . . . . . 9 ⊢ (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34		dfac12 10061	. . . . . . . . 9 ⊢ (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
35	33, 34	sylibr 234	. . . . . . . 8 ⊢ (ran ℵ ⊆ GCH → CHOICE)
36		dfac10 10049	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
37	35, 36	sylib 218	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → dom card = V)
38	16, 37	eleqtrrid 2842	. . . . . 6 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39		cardid2 9866	. . . . . 6 ⊢ (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40		engch 10540	. . . . . 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 2733	. 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 3049  Vcvv 3427   ∪ cun 3883   ⊆ wss 3885  𝒫 cpw 4531   class class class wbr 5074  dom cdm 5620  ran crn 5621  Oncon0 6312  suc csuc 6314   Fn wfn 6482  ‘cfv 6487  ωcom 7806   ≈ cen 8879  Fincfn 8882  cardccrd 9848  ℵcale 9849  CHOICEwac 10026  GCHcgch 10532

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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-reg 9496  ax-inf2 9551

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-isom 6496  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8100  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-seqom 8376  df-1o 8394  df-2o 8395  df-oadd 8398  df-omul 8399  df-oexp 8400  df-er 8632  df-map 8764  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-fsupp 9264  df-oi 9414  df-har 9461  df-wdom 9469  df-cnf 9572  df-r1 9677  df-rank 9678  df-dju 9814  df-card 9852  df-aleph 9853  df-ac 10027  df-fin4 10198  df-gch 10533

This theorem is referenced by:  gch3  10588