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Theorem gch2 10085
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
StepHypRef Expression
1 ssv 3988 . . 3 ran ℵ ⊆ V
2 sseq2 3990 . . 3 (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
31, 2mpbiri 259 . 2 (GCH = V → ran ℵ ⊆ GCH)
4 cardidm 9376 . . . . . . . 8 (card‘(card‘𝑥)) = (card‘𝑥)
5 iscard3 9507 . . . . . . . 8 ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
64, 5mpbi 231 . . . . . . 7 (card‘𝑥) ∈ (ω ∪ ran ℵ)
7 elun 4122 . . . . . . 7 ((card‘𝑥) ∈ (ω ∪ ran ℵ) ↔ ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ))
86, 7mpbi 231 . . . . . 6 ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ)
9 fingch 10033 . . . . . . . . 9 Fin ⊆ GCH
10 nnfi 8699 . . . . . . . . 9 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
119, 10sseldi 3962 . . . . . . . 8 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
1211a1i 11 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13 ssel 3958 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ran ℵ → (card‘𝑥) ∈ GCH))
1412, 13jaod 853 . . . . . 6 (ran ℵ ⊆ GCH → (((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ) → (card‘𝑥) ∈ GCH))
158, 14mpi 20 . . . . 5 (ran ℵ ⊆ GCH → (card‘𝑥) ∈ GCH)
16 vex 3495 . . . . . . 7 𝑥 ∈ V
17 alephon 9483 . . . . . . . . . . 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 9479 . . . . . . . . . . . . . 14 ℵ Fn On
21 fnfvelrn 6840 . . . . . . . . . . . . . 14 ((ℵ Fn On ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
2220, 18, 21sylancr 587 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
2319, 22sseldd 3965 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24 suceloni 7517 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → suc 𝑥 ∈ On)
2524adantl 482 . . . . . . . . . . . . . 14 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26 fnfvelrn 6840 . . . . . . . . . . . . . 14 ((ℵ Fn On ∧ suc 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
2720, 25, 26sylancr 587 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
2819, 27sseldd 3965 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29 gchaleph2 10082 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
3018, 23, 28, 29syl3anc 1363 . . . . . . . . . . 11 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
31 isnumi 9363 . . . . . . . . . . 11 (((ℵ‘suc 𝑥) ∈ On ∧ (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) → 𝒫 (ℵ‘𝑥) ∈ dom card)
3217, 30, 31sylancr 587 . . . . . . . . . 10 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝒫 (ℵ‘𝑥) ∈ dom card)
3332ralrimiva 3179 . . . . . . . . 9 (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34 dfac12 9563 . . . . . . . . 9 (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
3533, 34sylibr 235 . . . . . . . 8 (ran ℵ ⊆ GCH → CHOICE)
36 dfac10 9551 . . . . . . . 8 (CHOICE ↔ dom card = V)
3735, 36sylib 219 . . . . . . 7 (ran ℵ ⊆ GCH → dom card = V)
3816, 37eleqtrrid 2917 . . . . . 6 (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39 cardid2 9370 . . . . . 6 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40 engch 10038 . . . . . 6 ((card‘𝑥) ≈ 𝑥 → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
4138, 39, 403syl 18 . . . . 5 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
4215, 41mpbid 233 . . . 4 (ran ℵ ⊆ GCH → 𝑥 ∈ GCH)
4316a1i 11 . . . 4 (ran ℵ ⊆ GCH → 𝑥 ∈ V)
4442, 432thd 266 . . 3 (ran ℵ ⊆ GCH → (𝑥 ∈ GCH ↔ 𝑥 ∈ V))
4544eqrdv 2816 . 2 (ran ℵ ⊆ GCH → GCH = V)
463, 45impbii 210 1 (GCH = V ↔ ran ℵ ⊆ GCH)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cun 3931  wss 3933  𝒫 cpw 4535   class class class wbr 5057  dom cdm 5548  ran crn 5549  Oncon0 6184  suc csuc 6186   Fn wfn 6343  cfv 6348  ωcom 7569  cen 8494  Fincfn 8497  cardccrd 9352  cale 9353  CHOICEwac 9529  GCHcgch 10030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-reg 9044  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seqom 8073  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-oexp 8097  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-oi 8962  df-har 9010  df-wdom 9011  df-cnf 9113  df-r1 9181  df-rank 9182  df-dju 9318  df-card 9356  df-aleph 9357  df-ac 9530  df-fin4 9697  df-gch 10031
This theorem is referenced by:  gch3  10086
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