Theorem gch2 10558

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 3957	. . 3 ⊢ ran ℵ ⊆ V
2		sseq2 3959	. . 3 ⊢ (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
3	1, 2	mpbiri 258	. 2 ⊢ (GCH = V → ran ℵ ⊆ GCH)
4		cardidm 9844	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
5		iscard3 9976	. . . . . . . 8 ⊢ ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
6	4, 5	mpbi 230	. . . . . . 7 ⊢ (card‘𝑥) ∈ (ω ∪ ran ℵ)
7		elun 4101	. . . . . . 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 10506	. . . . . . . . 9 ⊢ Fin ⊆ GCH
10		nnfi 9072	. . . . . . . . 9 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
11	9, 10	sselid 3930	. . . . . . . 8 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
12	11	a1i 11	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13		ssel 3926	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ran ℵ → (card‘𝑥) ∈ GCH))
14	12, 13	jaod 859	. . . . . 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 3438	. . . . . . 7 ⊢ 𝑥 ∈ V
17		alephon 9952	. . . . . . . . . . 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 9948	. . . . . . . . . . . . . 14 ⊢ ℵ Fn On
21		fnfvelrn 7008	. . . . . . . . . . . . . 14 ⊢ ((ℵ Fn On ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
22	20, 18, 21	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
23	19, 22	sseldd 3933	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24		onsuc 7738	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
25	24	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26		fnfvelrn 7008	. . . . . . . . . . . . . 14 ⊢ ((ℵ Fn On ∧ suc 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
27	20, 25, 26	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
28	19, 27	sseldd 3933	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29		gchaleph2 10555	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
30	18, 23, 28, 29	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
31		isnumi 9831	. . . . . . . . . . 11 ⊢ (((ℵ‘suc 𝑥) ∈ On ∧ (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) → 𝒫 (ℵ‘𝑥) ∈ dom card)
32	17, 30, 31	sylancr 587	. . . . . . . . . 10 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝒫 (ℵ‘𝑥) ∈ dom card)
33	32	ralrimiva 3122	. . . . . . . . 9 ⊢ (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34		dfac12 10033	. . . . . . . . 9 ⊢ (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
35	33, 34	sylibr 234	. . . . . . . 8 ⊢ (ran ℵ ⊆ GCH → CHOICE)
36		dfac10 10021	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
37	35, 36	sylib 218	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → dom card = V)
38	16, 37	eleqtrrid 2836	. . . . . 6 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39		cardid2 9838	. . . . . 6 ⊢ (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40		engch 10511	. . . . . 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 2728	. 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 847   = wceq 1541   ∈ wcel 2110  ∀wral 3045  Vcvv 3434   ∪ cun 3898   ⊆ wss 3900  𝒫 cpw 4548   class class class wbr 5089  dom cdm 5614  ran crn 5615  Oncon0 6302  suc csuc 6304   Fn wfn 6472  ‘cfv 6477  ωcom 7791   ≈ cen 8861  Fincfn 8864  cardccrd 9820  ℵcale 9821  CHOICEwac 9998  GCHcgch 10503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-reg 9473  ax-inf2 9526

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-seqom 8362  df-1o 8380  df-2o 8381  df-oadd 8384  df-omul 8385  df-oexp 8386  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-oi 9391  df-har 9438  df-wdom 9446  df-cnf 9547  df-r1 9649  df-rank 9650  df-dju 9786  df-card 9824  df-aleph 9825  df-ac 9999  df-fin4 10170  df-gch 10504

This theorem is referenced by:  gch3  10559