Theorem gchaleph 10786

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 10220	. . 3 ⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)
2		alephon 10178	. . . . 5 ⊢ (ℵ‘suc 𝐴) ∈ On
3		onenon 10060	. . . . 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 10100	. . . 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 6924	. . . . . . 7 ⊢ (ℵ‘𝐴) ∈ V
10		simp1 1154	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐴) ∈ GCH ∧ 𝒫 (ℵ‘𝐴) ∈ dom card) → 𝐴 ∈ On)
11		alephgeom 10191	. . . . . . . 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 9058	. . . . . . 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 9153	. . . . . 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 9683	. . . . 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 10182	. . . . . 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 10739	. . . . . 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 10100	. . . 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 9147	. 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 2147  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560   class class class wbr 5106  dom cdm 5656  Oncon0 6383  suc csuc 6385  ‘cfv 6561  ωcom 7904   ≈ cen 9001   ≼ cdom 9002   ≺ csdm 9003  Fincfn 9004  cardccrd 10046  ℵcale 10047  GCHcgch 10735

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-om 7905  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-oi 9534  df-har 9581  df-card 10050  df-aleph 10051  df-gch 10736

This theorem is used by:  gchaleph2  10787