Theorem gch2 10424

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 3950	. . 3 ⊢ ran ℵ ⊆ V
2		sseq2 3952	. . 3 ⊢ (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
3	1, 2	mpbiri 257	. 2 ⊢ (GCH = V → ran ℵ ⊆ GCH)
4		cardidm 9710	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
5		iscard3 9842	. . . . . . . 8 ⊢ ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
6	4, 5	mpbi 229	. . . . . . 7 ⊢ (card‘𝑥) ∈ (ω ∪ ran ℵ)
7		elun 4088	. . . . . . 7 ⊢ ((card‘𝑥) ∈ (ω ∪ ran ℵ) ↔ ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ))
8	6, 7	mpbi 229	. . . . . 6 ⊢ ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ)
9		fingch 10372	. . . . . . . . 9 ⊢ Fin ⊆ GCH
10		nnfi 8924	. . . . . . . . 9 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
11	9, 10	sselid 3924	. . . . . . . 8 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
12	11	a1i 11	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13		ssel 3919	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ran ℵ → (card‘𝑥) ∈ GCH))
14	12, 13	jaod 856	. . . . . 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 3435	. . . . . . 7 ⊢ 𝑥 ∈ V
17		alephon 9818	. . . . . . . . . . 11 ⊢ (ℵ‘suc 𝑥) ∈ On
18		simpr 485	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
19		simpl 483	. . . . . . . . . . . . 13 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → ran ℵ ⊆ GCH)
20		alephfnon 9814	. . . . . . . . . . . . . 14 ⊢ ℵ Fn On
21		fnfvelrn 6953	. . . . . . . . . . . . . 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 3927	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24		suceloni 7649	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
25	24	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26		fnfvelrn 6953	. . . . . . . . . . . . . 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 3927	. . . . . . . . . . . 12 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29		gchaleph2 10421	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
30	18, 23, 28, 29	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
31		isnumi 9697	. . . . . . . . . . 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 3110	. . . . . . . . 9 ⊢ (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34		dfac12 9898	. . . . . . . . 9 ⊢ (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
35	33, 34	sylibr 233	. . . . . . . 8 ⊢ (ran ℵ ⊆ GCH → CHOICE)
36		dfac10 9886	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
37	35, 36	sylib 217	. . . . . . 7 ⊢ (ran ℵ ⊆ GCH → dom card = V)
38	16, 37	eleqtrrid 2848	. . . . . 6 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39		cardid2 9704	. . . . . 6 ⊢ (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40		engch 10377	. . . . . 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 231	. . . 4 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ GCH)
43	16	a1i 11	. . . 4 ⊢ (ran ℵ ⊆ GCH → 𝑥 ∈ V)
44	42, 43	2thd 264	. . 3 ⊢ (ran ℵ ⊆ GCH → (𝑥 ∈ GCH ↔ 𝑥 ∈ V))
45	44	eqrdv 2738	. 2 ⊢ (ran ℵ ⊆ GCH → GCH = V)
46	3, 45	impbii 208	1 ⊢ (GCH = V ↔ ran ℵ ⊆ GCH)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1542   ∈ wcel 2110  ∀wral 3066  Vcvv 3431   ∪ cun 3890   ⊆ wss 3892  𝒫 cpw 4539   class class class wbr 5079  dom cdm 5589  ran crn 5590  Oncon0 6264  suc csuc 6266   Fn wfn 6426  ‘cfv 6431  ωcom 7701   ≈ cen 8705  Fincfn 8708  cardccrd 9686  ℵcale 9687  CHOICEwac 9864  GCHcgch 10369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-reg 9321  ax-inf2 9369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-isom 6440  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7702  df-1st 7818  df-2nd 7819  df-supp 7963  df-frecs 8082  df-wrecs 8113  df-recs 8187  df-rdg 8226  df-seqom 8264  df-1o 8282  df-2o 8283  df-oadd 8286  df-omul 8287  df-oexp 8288  df-er 8473  df-map 8592  df-en 8709  df-dom 8710  df-sdom 8711  df-fin 8712  df-fsupp 9099  df-oi 9239  df-har 9286  df-wdom 9294  df-cnf 9390  df-r1 9515  df-rank 9516  df-dju 9652  df-card 9690  df-aleph 9691  df-ac 9865  df-fin4 10036  df-gch 10370

This theorem is referenced by:  gch3  10425