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Theorem gchaleph2 10083
Description: If (ℵ‘𝐴) and (ℵ‘suc 𝐴) are GCH-sets, then the successor aleph (ℵ‘suc 𝐴) is equinumerous to the powerset of (ℵ‘𝐴). (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
gchaleph2 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))

Proof of Theorem gchaleph2
StepHypRef Expression
1 harcl 9011 . . 3 (har‘(ℵ‘𝐴)) ∈ On
2 alephon 9484 . . . . 5 (ℵ‘𝐴) ∈ On
3 onenon 9366 . . . . 5 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
4 harsdom 9412 . . . . 5 ((ℵ‘𝐴) ∈ dom card → (ℵ‘𝐴) ≺ (har‘(ℵ‘𝐴)))
52, 3, 4mp2b 10 . . . 4 (ℵ‘𝐴) ≺ (har‘(ℵ‘𝐴))
6 simp1 1133 . . . . . . 7 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → 𝐴 ∈ On)
7 alephgeom 9497 . . . . . . 7 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
86, 7sylib 221 . . . . . 6 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → ω ⊆ (ℵ‘𝐴))
9 ssdomg 8542 . . . . . 6 ((ℵ‘𝐴) ∈ On → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
102, 8, 9mpsyl 68 . . . . 5 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → ω ≼ (ℵ‘𝐴))
11 simp2 1134 . . . . 5 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (ℵ‘𝐴) ∈ GCH)
12 alephsuc 9483 . . . . . . 7 (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
136, 12syl 17 . . . . . 6 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴)))
14 simp3 1135 . . . . . 6 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (ℵ‘suc 𝐴) ∈ GCH)
1513, 14eqeltrrd 2915 . . . . 5 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (har‘(ℵ‘𝐴)) ∈ GCH)
16 gchpwdom 10081 . . . . 5 ((ω ≼ (ℵ‘𝐴) ∧ (ℵ‘𝐴) ∈ GCH ∧ (har‘(ℵ‘𝐴)) ∈ GCH) → ((ℵ‘𝐴) ≺ (har‘(ℵ‘𝐴)) ↔ 𝒫 (ℵ‘𝐴) ≼ (har‘(ℵ‘𝐴))))
1710, 11, 15, 16syl3anc 1368 . . . 4 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → ((ℵ‘𝐴) ≺ (har‘(ℵ‘𝐴)) ↔ 𝒫 (ℵ‘𝐴) ≼ (har‘(ℵ‘𝐴))))
185, 17mpbii 236 . . 3 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → 𝒫 (ℵ‘𝐴) ≼ (har‘(ℵ‘𝐴)))
19 ondomen 9452 . . 3 (((har‘(ℵ‘𝐴)) ∈ On ∧ 𝒫 (ℵ‘𝐴) ≼ (har‘(ℵ‘𝐴))) → 𝒫 (ℵ‘𝐴) ∈ dom card)
201, 18, 19sylancr 590 . 2 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → 𝒫 (ℵ‘𝐴) ∈ dom card)
21 gchaleph 10082 . 2 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))
2220, 21syld3an3 1406 1 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2114  wss 3908  𝒫 cpw 4511   class class class wbr 5042  dom cdm 5532  Oncon0 6169  suc csuc 6171  cfv 6334  ωcom 7565  cen 8493  cdom 8494  csdm 8495  harchar 9008  cardccrd 9352  cale 9353  GCHcgch 10031
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-oexp 8095  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-har 9009  df-wdom 9017  df-cnf 9113  df-dju 9318  df-card 9356  df-aleph 9357  df-fin4 9698  df-gch 10032
This theorem is referenced by:  gch2  10086  gch3  10087
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