Theorem gchaleph 10427

Description: If (ℵ‘𝐴) is a GCH-set and its powerset is well-orderable, then the successor aleph (ℵ‘suc 𝐴) is equinumerous to the powerset of (ℵ‘𝐴). (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
gchaleph	⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))

Proof of Theorem gchaleph

Step	Hyp	Ref	Expression
1		alephsucpw2 9867	. . 3 ⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)
2		alephon 9825	. . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On
3		onenon 9707	. . . . 5 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
4	2, 3	ax-mp 5	. . . 4 ⊢ (ℵ‘suc 𝐴) ∈ dom card
5		simp3 1137	. . . 4 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝒫 (ℵ‘𝐴) ∈ dom card)
6		domtri2 9747	. . . 4 ⊢ (((ℵ‘suc 𝐴) ∈ dom card ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ((ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) ↔ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)))
7	4, 5, 6	sylancr 587	. . 3 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ((ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) ↔ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)))
8	1, 7	mpbiri 257	. 2 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴))
9		fvex 6787	. . . . . . 7 ⊢ (ℵ‘𝐴) ∈ V
10		simp1 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝐴 ∈ On)
11		alephgeom 9838	. . . . . . . 8 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
12	10, 11	sylib 217	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ω ⊆ (ℵ‘𝐴))
13		ssdomg 8786	. . . . . . 7 ⊢ ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
14	9, 12, 13	mpsyl 68	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ω ≼ (ℵ‘𝐴))
15		domnsym 8886	. . . . . 6 ⊢ (ω ≼ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≺ ω)
16	14, 15	syl 17	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ¬ (ℵ‘𝐴) ≺ ω)
17		isfinite 9410	. . . . 5 ⊢ ((ℵ‘𝐴) ∈ Fin ↔ (ℵ‘𝐴) ≺ ω)
18	16, 17	sylnibr 329	. . . 4 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ¬ (ℵ‘𝐴) ∈ Fin)
19		simp2 1136	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘𝐴) ∈ GCH)
20		alephordilem1 9829	. . . . . 6 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
21	20	3ad2ant1 1132	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
22		gchi 10380	. . . . . 6 ⊢ (((ℵ‘𝐴) ∈ GCH ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) ∧ (ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴)) → (ℵ‘𝐴) ∈ Fin)
23	22	3expia 1120	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ GCH ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → ((ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴) → (ℵ‘𝐴) ∈ Fin))
24	19, 21, 23	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ((ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴) → (ℵ‘𝐴) ∈ Fin))
25	18, 24	mtod 197	. . 3 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ¬ (ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴))
26		domtri2 9747	. . . 4 ⊢ ((𝒫 (ℵ‘𝐴) ∈ dom card ∧ (ℵ‘suc 𝐴) ∈ dom card) → (𝒫 (ℵ‘𝐴) ≼ (ℵ‘suc 𝐴) ↔ ¬ (ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴)))
27	5, 4, 26	sylancl 586	. . 3 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (𝒫 (ℵ‘𝐴) ≼ (ℵ‘suc 𝐴) ↔ ¬ (ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴)))
28	25, 27	mpbird 256	. 2 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝒫 (ℵ‘𝐴) ≼ (ℵ‘suc 𝐴))
29		sbth 8880	. 2 ⊢ (((ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≼ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))
30	8, 28, 29	syl2anc 584	1 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ w3a 1086   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533   class class class wbr 5074  dom cdm 5589  Oncon0 6266  suc csuc 6268  ‘cfv 6433  ωcom 7712   ≈ cen 8730   ≼ cdom 8731   ≺ csdm 8732  Fincfn 8733  cardccrd 9693  ℵcale 9694  GCHcgch 10376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  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 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-oi 9269  df-har 9316  df-card 9697  df-aleph 9698  df-gch 10377

This theorem is referenced by:  gchaleph2  10428