Theorem gchaleph 10616

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 10056	. . 3 ⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)
2		alephon 10014	. . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On
3		onenon 9894	. . . . 5 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
4	2, 3	ax-mp 5	. . . 4 ⊢ (ℵ‘suc 𝐴) ∈ dom card
5		simp3 1138	. . . 4 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝒫 (ℵ‘𝐴) ∈ dom card)
6		domtri2 9934	. . . 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 6860	. . . . . . 7 ⊢ (ℵ‘𝐴) ∈ V
10		simp1 1136	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝐴 ∈ On)
11		alephgeom 10027	. . . . . . . 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 8947	. . . . . . 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 9050	. . . . . 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 9597	. . . . 5 ⊢ ((ℵ‘𝐴) ∈ Fin ↔ (ℵ‘𝐴) ≺ ω)
18	16, 17	sylnibr 328	. . . 4 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ¬ (ℵ‘𝐴) ∈ Fin)
19		simp2 1137	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘𝐴) ∈ GCH)
20		alephordilem1 10018	. . . . . 6 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
21	20	3ad2ant1 1133	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
22		gchi 10569	. . . . . 6 ⊢ (((ℵ‘𝐴) ∈ GCH ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) ∧ (ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴)) → (ℵ‘𝐴) ∈ Fin)
23	22	3expia 1121	. . . . 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 9934	. . . 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 9044	. 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 1087   ∈ wcel 2106  Vcvv 3446   ⊆ wss 3913  𝒫 cpw 4565   class class class wbr 5110  dom cdm 5638  Oncon0 6322  suc csuc 6324  ‘cfv 6501  ωcom 7807   ≈ cen 8887   ≼ cdom 8888   ≺ csdm 8889  Fincfn 8890  cardccrd 9880  ℵcale 9881  GCHcgch 10565

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9455  df-har 9502  df-card 9884  df-aleph 9885  df-gch 10566

This theorem is referenced by:  gchaleph2  10617