Theorem gchaleph 10358

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 9798	. . 3 ⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)
2		alephon 9756	. . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On
3		onenon 9638	. . . . 5 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
4	2, 3	ax-mp 5	. . . 4 ⊢ (ℵ‘suc 𝐴) ∈ dom card
5		simp3 1136	. . . 4 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝒫 (ℵ‘𝐴) ∈ dom card)
6		domtri2 9678	. . . 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 586	. . 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 6769	. . . . . . 7 ⊢ (ℵ‘𝐴) ∈ V
10		simp1 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝐴 ∈ On)
11		alephgeom 9769	. . . . . . . 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 8741	. . . . . . 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 8839	. . . . . 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 9340	. . . . 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 1135	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘𝐴) ∈ GCH)
20		alephordilem1 9760	. . . . . 6 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
21	20	3ad2ant1 1131	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
22		gchi 10311	. . . . . 6 ⊢ (((ℵ‘𝐴) ∈ GCH ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) ∧ (ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴)) → (ℵ‘𝐴) ∈ Fin)
23	22	3expia 1119	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ GCH ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → ((ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴) → (ℵ‘𝐴) ∈ Fin))
24	19, 21, 23	syl2anc 583	. . . 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 9678	. . . 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 585	. . 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 8833	. 2 ⊢ (((ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≼ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))
30	8, 28, 29	syl2anc 583	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 1085   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070  dom cdm 5580  Oncon0 6251  suc csuc 6253  ‘cfv 6418  ωcom 7687   ≈ cen 8688   ≼ cdom 8689   ≺ csdm 8690  Fincfn 8691  cardccrd 9624  ℵcale 9625  GCHcgch 10307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-har 9246  df-card 9628  df-aleph 9629  df-gch 10308

This theorem is referenced by:  gchaleph2  10359