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Theorem gchaleph2 9892
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 8820 . . 3 (har‘(ℵ‘𝐴)) ∈ On
2 alephon 9289 . . . . 5 (ℵ‘𝐴) ∈ On
3 onenon 9172 . . . . 5 ((ℵ‘𝐴) ∈ On → (ℵ‘𝐴) ∈ dom card)
4 harsdom 9218 . . . . 5 ((ℵ‘𝐴) ∈ dom card → (ℵ‘𝐴) ≺ (har‘(ℵ‘𝐴)))
52, 3, 4mp2b 10 . . . 4 (ℵ‘𝐴) ≺ (har‘(ℵ‘𝐴))
6 simp1 1116 . . . . . . 7 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → 𝐴 ∈ On)
7 alephgeom 9302 . . . . . . 7 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
86, 7sylib 210 . . . . . 6 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → ω ⊆ (ℵ‘𝐴))
9 ssdomg 8352 . . . . . 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 1117 . . . . 5 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (ℵ‘𝐴) ∈ GCH)
12 alephsuc 9288 . . . . . . 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 1118 . . . . . 6 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (ℵ‘suc 𝐴) ∈ GCH)
1513, 14eqeltrrd 2867 . . . . 5 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → (har‘(ℵ‘𝐴)) ∈ GCH)
16 gchpwdom 9890 . . . . 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 1351 . . . 4 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → ((ℵ‘𝐴) ≺ (har‘(ℵ‘𝐴)) ↔ 𝒫 (ℵ‘𝐴) ≼ (har‘(ℵ‘𝐴))))
185, 17mpbii 225 . . 3 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → 𝒫 (ℵ‘𝐴) ≼ (har‘(ℵ‘𝐴)))
19 ondomen 9257 . . 3 (((har‘(ℵ‘𝐴)) ∈ On ∧ 𝒫 (ℵ‘𝐴) ≼ (har‘(ℵ‘𝐴))) → 𝒫 (ℵ‘𝐴) ∈ dom card)
201, 18, 19sylancr 578 . 2 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ (ℵ‘suc 𝐴) ∈ GCH) → 𝒫 (ℵ‘𝐴) ∈ dom card)
21 gchaleph 9891 . 2 ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))
2220, 21syld3an3 1389 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 198  w3a 1068   = wceq 1507  wcel 2050  wss 3829  𝒫 cpw 4422   class class class wbr 4929  dom cdm 5407  Oncon0 6029  suc csuc 6031  cfv 6188  ωcom 7396  cen 8303  cdom 8304  csdm 8305  harchar 8815  cardccrd 9158  cale 9159  GCHcgch 9840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-supp 7634  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-seqom 7887  df-1o 7905  df-2o 7906  df-oadd 7909  df-omul 7910  df-oexp 7911  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-fsupp 8629  df-oi 8769  df-har 8817  df-wdom 8818  df-cnf 8919  df-dju 9124  df-card 9162  df-aleph 9163  df-fin4 9507  df-gch 9841
This theorem is referenced by:  gch2  9895  gch3  9896
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