Theorem gchhar 9380

Description: A "local" form of gchac 9382. If 𝐴 and 𝒫 𝐴 are GCH-sets, then the Hartogs number of 𝐴 is 𝒫 𝐴 (so 𝒫 𝐴 and a fortiori 𝐴 are well-orderable). The proof is due to Specker. Theorem 2.1 of [KanamoriPincus] p. 419. (Contributed by Mario Carneiro, 31-May-2015.)

Assertion

Ref	Expression
gchhar	⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≈ 𝒫 𝐴)

Proof of Theorem gchhar

Step	Hyp	Ref	Expression
1		harcl 8349	. . . 4 ⊢ (har‘𝐴) ∈ On
2		simp3 1056	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ∈ GCH)
3		cdadom3 8893	. . . 4 ⊢ (((har‘𝐴) ∈ On ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝒫 𝐴))
4	1, 2, 3	sylancr 694	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝒫 𝐴))
5		domnsym 7971	. . . . . . . . 9 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ≺ ω)
6	5	3ad2ant1 1075	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ¬ 𝐴 ≺ ω)
7		isfinite 8432	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
8	6, 7	sylnibr 318	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ¬ 𝐴 ∈ Fin)
9		pwfi 8144	. . . . . . 7 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
10	8, 9	sylnib 317	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ¬ 𝒫 𝐴 ∈ Fin)
11		cdadom3 8893	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ GCH ∧ (har‘𝐴) ∈ On) → 𝒫 𝐴 ≼ (𝒫 𝐴 +𝑐 (har‘𝐴)))
12	2, 1, 11	sylancl 693	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≼ (𝒫 𝐴 +𝑐 (har‘𝐴)))
13		ovex 6577	. . . . . . . 8 ⊢ (𝒫 𝐴 +𝑐 (har‘𝐴)) ∈ V
14	13	canth2 7998	. . . . . . 7 ⊢ (𝒫 𝐴 +𝑐 (har‘𝐴)) ≺ 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴))
15		pwcdaen 8890	. . . . . . . . 9 ⊢ ((𝒫 𝐴 ∈ GCH ∧ (har‘𝐴) ∈ On) → 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ≈ (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)))
16	2, 1, 15	sylancl 693	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ≈ (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)))
17		pwexg 4776	. . . . . . . . . . 11 ⊢ (𝒫 𝐴 ∈ GCH → 𝒫 𝒫 𝐴 ∈ V)
18	2, 17	syl 17	. . . . . . . . . 10 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝒫 𝐴 ∈ V)
19		simp2 1055	. . . . . . . . . . 11 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝐴 ∈ GCH)
20		harwdom 8378	. . . . . . . . . . 11 ⊢ (𝐴 ∈ GCH → (har‘𝐴) ≼* 𝒫 (𝐴 × 𝐴))
21		wdompwdom 8366	. . . . . . . . . . 11 ⊢ ((har‘𝐴) ≼* 𝒫 (𝐴 × 𝐴) → 𝒫 (har‘𝐴) ≼ 𝒫 𝒫 (𝐴 × 𝐴))
22	19, 20, 21	3syl 18	. . . . . . . . . 10 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (har‘𝐴) ≼ 𝒫 𝒫 (𝐴 × 𝐴))
23		xpdom2g 7941	. . . . . . . . . 10 ⊢ ((𝒫 𝒫 𝐴 ∈ V ∧ 𝒫 (har‘𝐴) ≼ 𝒫 𝒫 (𝐴 × 𝐴)) → (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)) ≼ (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)))
24	18, 22, 23	syl2anc 691	. . . . . . . . 9 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)) ≼ (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)))
25		xpexg 6858	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ GCH ∧ 𝐴 ∈ GCH) → (𝐴 × 𝐴) ∈ V)
26	19, 19, 25	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 × 𝐴) ∈ V)
27		pwexg 4776	. . . . . . . . . . . . 13 ⊢ ((𝐴 × 𝐴) ∈ V → 𝒫 (𝐴 × 𝐴) ∈ V)
28	26, 27	syl 17	. . . . . . . . . . . 12 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝐴 × 𝐴) ∈ V)
29		pwcdaen 8890	. . . . . . . . . . . 12 ⊢ ((𝒫 𝐴 ∈ GCH ∧ 𝒫 (𝐴 × 𝐴) ∈ V) → 𝒫 (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)))
30	2, 28, 29	syl2anc 691	. . . . . . . . . . 11 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)))
31	30	ensymd 7893	. . . . . . . . . 10 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)))
32		enrefg 7873	. . . . . . . . . . . . . 14 ⊢ (𝒫 𝐴 ∈ GCH → 𝒫 𝐴 ≈ 𝒫 𝐴)
33	2, 32	syl 17	. . . . . . . . . . . . 13 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≈ 𝒫 𝐴)
34		gchxpidm 9370	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 × 𝐴) ≈ 𝐴)
35	19, 8, 34	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 × 𝐴) ≈ 𝐴)
36		pwen 8018	. . . . . . . . . . . . . 14 ⊢ ((𝐴 × 𝐴) ≈ 𝐴 → 𝒫 (𝐴 × 𝐴) ≈ 𝒫 𝐴)
37	35, 36	syl 17	. . . . . . . . . . . . 13 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝐴 × 𝐴) ≈ 𝒫 𝐴)
38		cdaen 8878	. . . . . . . . . . . . 13 ⊢ ((𝒫 𝐴 ≈ 𝒫 𝐴 ∧ 𝒫 (𝐴 × 𝐴) ≈ 𝒫 𝐴) → (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴))
39	33, 37, 38	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴))
40		gchcdaidm 9369	. . . . . . . . . . . . 13 ⊢ ((𝒫 𝐴 ∈ GCH ∧ ¬ 𝒫 𝐴 ∈ Fin) → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴)
41	2, 10, 40	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴)
42		entr 7894	. . . . . . . . . . . 12 ⊢ (((𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴) ∧ (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴) → (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝐴)
43	39, 41, 42	syl2anc 691	. . . . . . . . . . 11 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝐴)
44		pwen 8018	. . . . . . . . . . 11 ⊢ ((𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝐴 → 𝒫 (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝒫 𝐴)
45	43, 44	syl 17	. . . . . . . . . 10 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝒫 𝐴)
46		entr 7894	. . . . . . . . . 10 ⊢ (((𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ∧ 𝒫 (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝒫 𝐴) → (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝒫 𝐴)
47	31, 45, 46	syl2anc 691	. . . . . . . . 9 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝒫 𝐴)
48		domentr 7901	. . . . . . . . 9 ⊢ (((𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)) ≼ (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)) ∧ (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝒫 𝐴) → (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)) ≼ 𝒫 𝒫 𝐴)
49	24, 47, 48	syl2anc 691	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)) ≼ 𝒫 𝒫 𝐴)
50		endomtr 7900	. . . . . . . 8 ⊢ ((𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ≈ (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)) ∧ (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)) ≼ 𝒫 𝒫 𝐴) → 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ≼ 𝒫 𝒫 𝐴)
51	16, 49, 50	syl2anc 691	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ≼ 𝒫 𝒫 𝐴)
52		sdomdomtr 7978	. . . . . . 7 ⊢ (((𝒫 𝐴 +𝑐 (har‘𝐴)) ≺ 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ∧ 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ≼ 𝒫 𝒫 𝐴) → (𝒫 𝐴 +𝑐 (har‘𝐴)) ≺ 𝒫 𝒫 𝐴)
53	14, 51, 52	sylancr 694	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝐴 +𝑐 (har‘𝐴)) ≺ 𝒫 𝒫 𝐴)
54		gchen1 9326	. . . . . 6 ⊢ (((𝒫 𝐴 ∈ GCH ∧ ¬ 𝒫 𝐴 ∈ Fin) ∧ (𝒫 𝐴 ≼ (𝒫 𝐴 +𝑐 (har‘𝐴)) ∧ (𝒫 𝐴 +𝑐 (har‘𝐴)) ≺ 𝒫 𝒫 𝐴)) → 𝒫 𝐴 ≈ (𝒫 𝐴 +𝑐 (har‘𝐴)))
55	2, 10, 12, 53, 54	syl22anc 1319	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≈ (𝒫 𝐴 +𝑐 (har‘𝐴)))
56		cdacomen 8886	. . . . 5 ⊢ (𝒫 𝐴 +𝑐 (har‘𝐴)) ≈ ((har‘𝐴) +𝑐 𝒫 𝐴)
57		entr 7894	. . . . 5 ⊢ ((𝒫 𝐴 ≈ (𝒫 𝐴 +𝑐 (har‘𝐴)) ∧ (𝒫 𝐴 +𝑐 (har‘𝐴)) ≈ ((har‘𝐴) +𝑐 𝒫 𝐴)) → 𝒫 𝐴 ≈ ((har‘𝐴) +𝑐 𝒫 𝐴))
58	55, 56, 57	sylancl 693	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≈ ((har‘𝐴) +𝑐 𝒫 𝐴))
59	58	ensymd 7893	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ((har‘𝐴) +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴)
60		domentr 7901	. . 3 ⊢ (((har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝒫 𝐴) ∧ ((har‘𝐴) +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴) → (har‘𝐴) ≼ 𝒫 𝐴)
61	4, 59, 60	syl2anc 691	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ 𝒫 𝐴)
62		gchcdaidm 9369	. . . . . 6 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 +𝑐 𝐴) ≈ 𝐴)
63	19, 8, 62	syl2anc 691	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 +𝑐 𝐴) ≈ 𝐴)
64		pwen 8018	. . . . 5 ⊢ ((𝐴 +𝑐 𝐴) ≈ 𝐴 → 𝒫 (𝐴 +𝑐 𝐴) ≈ 𝒫 𝐴)
65	63, 64	syl 17	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝐴 +𝑐 𝐴) ≈ 𝒫 𝐴)
66		cdadom3 8893	. . . . . . . 8 ⊢ ((𝐴 ∈ GCH ∧ (har‘𝐴) ∈ On) → 𝐴 ≼ (𝐴 +𝑐 (har‘𝐴)))
67	19, 1, 66	sylancl 693	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝐴 ≼ (𝐴 +𝑐 (har‘𝐴)))
68		harndom 8352	. . . . . . . 8 ⊢ ¬ (har‘𝐴) ≼ 𝐴
69		cdadom3 8893	. . . . . . . . . . 11 ⊢ (((har‘𝐴) ∈ On ∧ 𝐴 ∈ GCH) → (har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝐴))
70	1, 19, 69	sylancr 694	. . . . . . . . . 10 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝐴))
71		cdacomen 8886	. . . . . . . . . 10 ⊢ ((har‘𝐴) +𝑐 𝐴) ≈ (𝐴 +𝑐 (har‘𝐴))
72		domentr 7901	. . . . . . . . . 10 ⊢ (((har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝐴) ∧ ((har‘𝐴) +𝑐 𝐴) ≈ (𝐴 +𝑐 (har‘𝐴))) → (har‘𝐴) ≼ (𝐴 +𝑐 (har‘𝐴)))
73	70, 71, 72	sylancl 693	. . . . . . . . 9 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ (𝐴 +𝑐 (har‘𝐴)))
74		domen2 7988	. . . . . . . . 9 ⊢ (𝐴 ≈ (𝐴 +𝑐 (har‘𝐴)) → ((har‘𝐴) ≼ 𝐴 ↔ (har‘𝐴) ≼ (𝐴 +𝑐 (har‘𝐴))))
75	73, 74	syl5ibrcom 236	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 ≈ (𝐴 +𝑐 (har‘𝐴)) → (har‘𝐴) ≼ 𝐴))
76	68, 75	mtoi 189	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ¬ 𝐴 ≈ (𝐴 +𝑐 (har‘𝐴)))
77		brsdom 7864	. . . . . . 7 ⊢ (𝐴 ≺ (𝐴 +𝑐 (har‘𝐴)) ↔ (𝐴 ≼ (𝐴 +𝑐 (har‘𝐴)) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 (har‘𝐴))))
78	67, 76, 77	sylanbrc 695	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝐴 ≺ (𝐴 +𝑐 (har‘𝐴)))
79		canth2g 7999	. . . . . . . . 9 ⊢ (𝐴 ∈ GCH → 𝐴 ≺ 𝒫 𝐴)
80		sdomdom 7869	. . . . . . . . 9 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴)
81		cdadom1 8891	. . . . . . . . 9 ⊢ (𝐴 ≼ 𝒫 𝐴 → (𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 (har‘𝐴)))
82	19, 79, 80, 81	4syl 19	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 (har‘𝐴)))
83		cdadom2 8892	. . . . . . . . 9 ⊢ ((har‘𝐴) ≼ 𝒫 𝐴 → (𝒫 𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 𝒫 𝐴))
84	61, 83	syl 17	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 𝒫 𝐴))
85		domtr 7895	. . . . . . . 8 ⊢ (((𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 (har‘𝐴)) ∧ (𝒫 𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 𝒫 𝐴)) → (𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 𝒫 𝐴))
86	82, 84, 85	syl2anc 691	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 𝒫 𝐴))
87		domentr 7901	. . . . . . 7 ⊢ (((𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 𝒫 𝐴) ∧ (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴) → (𝐴 +𝑐 (har‘𝐴)) ≼ 𝒫 𝐴)
88	86, 41, 87	syl2anc 691	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 +𝑐 (har‘𝐴)) ≼ 𝒫 𝐴)
89		gchen2 9327	. . . . . 6 ⊢ (((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴 ≺ (𝐴 +𝑐 (har‘𝐴)) ∧ (𝐴 +𝑐 (har‘𝐴)) ≼ 𝒫 𝐴)) → (𝐴 +𝑐 (har‘𝐴)) ≈ 𝒫 𝐴)
90	19, 8, 78, 88, 89	syl22anc 1319	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 +𝑐 (har‘𝐴)) ≈ 𝒫 𝐴)
91	90	ensymd 7893	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≈ (𝐴 +𝑐 (har‘𝐴)))
92		entr 7894	. . . 4 ⊢ ((𝒫 (𝐴 +𝑐 𝐴) ≈ 𝒫 𝐴 ∧ 𝒫 𝐴 ≈ (𝐴 +𝑐 (har‘𝐴))) → 𝒫 (𝐴 +𝑐 𝐴) ≈ (𝐴 +𝑐 (har‘𝐴)))
93	65, 91, 92	syl2anc 691	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝐴 +𝑐 𝐴) ≈ (𝐴 +𝑐 (har‘𝐴)))
94		endom 7868	. . 3 ⊢ (𝒫 (𝐴 +𝑐 𝐴) ≈ (𝐴 +𝑐 (har‘𝐴)) → 𝒫 (𝐴 +𝑐 𝐴) ≼ (𝐴 +𝑐 (har‘𝐴)))
95		pwcdadom 8921	. . 3 ⊢ (𝒫 (𝐴 +𝑐 𝐴) ≼ (𝐴 +𝑐 (har‘𝐴)) → 𝒫 𝐴 ≼ (har‘𝐴))
96	93, 94, 95	3syl 18	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≼ (har‘𝐴))
97		sbth 7965	. 2 ⊢ (((har‘𝐴) ≼ 𝒫 𝐴 ∧ 𝒫 𝐴 ≼ (har‘𝐴)) → (har‘𝐴) ≈ 𝒫 𝐴)
98	61, 96, 97	syl2anc 691	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≈ 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1031   ∈ wcel 1977  Vcvv 3173  𝒫 cpw 4108   class class class wbr 4583   × cxp 5036  Oncon0 5640  ‘cfv 5804  (class class class)co 6549  ωcom 6957   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841  harchar 8344   ≼^* cwdom 8345   +_𝑐 ccda 8872  GCHcgch 9321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-har 8346  df-wdom 8347  df-cnf 8442  df-card 8648  df-cda 8873  df-fin4 8992  df-gch 9322

This theorem is referenced by:  gchacg  9381