Theorem gchaleph 10759

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 10193	. . 3 ⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)
2		alephon 10151	. . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On
3		onenon 10033	. . . . 5 ⊢ ((ℵ‘suc 𝐴) ∈ On → (ℵ‘suc 𝐴) ∈ dom card)
4	2, 3	ax-mp 5	. . . 4 ⊢ (ℵ‘suc 𝐴) ∈ dom card
5		simp3 1156	. . . 4 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝒫 (ℵ‘𝐴) ∈ dom card)
6		domtri2 10073	. . . 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 599	. . 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 261	. 2 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴))
9		fvex 6898	. . . . . . 7 ⊢ (ℵ‘𝐴) ∈ V
10		simp1 1154	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝐴 ∈ On)
11		alephgeom 10164	. . . . . . . 8 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
12	10, 11	sylib 221	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ω ⊆ (ℵ‘𝐴))
13		ssdomg 9030	. . . . . . 7 ⊢ ((ℵ‘𝐴) ∈ V → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
14	9, 12, 13	mpsyl 69	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ω ≼ (ℵ‘𝐴))
15		domnsym 9125	. . . . . 6 ⊢ (ω ≼ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≺ ω)
16	14, 15	syl 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ¬ (ℵ‘𝐴) ≺ ω)
17		isfinite 9656	. . . . 5 ⊢ ((ℵ‘𝐴) ∈ Fin ↔ (ℵ‘𝐴) ≺ ω)
18	16, 17	sylnibr 332	. . . 4 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ¬ (ℵ‘𝐴) ∈ Fin)
19		simp2 1155	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘𝐴) ∈ GCH)
20		alephordilem1 10155	. . . . . 6 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
21	20	3ad2ant1 1151	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
22		gchi 10712	. . . . . 6 ⊢ (((ℵ‘𝐴) ∈ GCH ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) ∧ (ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴)) → (ℵ‘𝐴) ∈ Fin)
23	22	3expia 1139	. . . . 5 ⊢ (((ℵ‘𝐴) ∈ GCH ∧ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) → ((ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴) → (ℵ‘𝐴) ∈ Fin))
24	19, 21, 23	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ((ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴) → (ℵ‘𝐴) ∈ Fin))
25	18, 24	mtod 201	. . 3 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → ¬ (ℵ‘suc 𝐴) ≺ 𝒫 (ℵ‘𝐴))
26		domtri2 10073	. . . 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 598	. . 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 260	. 2 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝒫 (ℵ‘𝐴) ≼ (ℵ‘suc 𝐴))
29		sbth 9119	. 2 ⊢ (((ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≼ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))
30	8, 28, 29	syl2anc 596	1 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → (ℵ‘suc 𝐴) ≈ 𝒫 (ℵ‘𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ w3a 1103   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895  𝒫 cpw 4556   class class class wbr 5102  dom cdm 5650  Oncon0 6362  suc csuc 6364  ‘cfv 6538  ωcom 7877   ≈ cen 8973   ≼ cdom 8974   ≺ csdm 8975  Fincfn 8976  cardccrd 10019  ℵcale 10020  GCHcgch 10708

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9645

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-om 7878  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-oi 9507  df-har 9554  df-card 10023  df-aleph 10024  df-gch 10709

This theorem is used by:  gchaleph2  10760