Theorem gchhar 9501

Description: A "local" form of gchac 9503. 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 8466	. . . 4 ⊢ (har‘𝐴) ∈ On
2		simp3 1063	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ∈ GCH)
3		cdadom3 9010	. . . 4 ⊢ (((har‘𝐴) ∈ On ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝒫 𝐴))
4	1, 2, 3	sylancr 695	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝒫 𝐴))
5		domnsym 8086	. . . . . . . . 9 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ≺ ω)
6	5	3ad2ant1 1082	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ¬ 𝐴 ≺ ω)
7		isfinite 8549	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
8	6, 7	sylnibr 319	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ¬ 𝐴 ∈ Fin)
9		pwfi 8261	. . . . . . 7 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
10	8, 9	sylnib 318	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ¬ 𝒫 𝐴 ∈ Fin)
11		cdadom3 9010	. . . . . . 7 ⊢ ((𝒫 𝐴 ∈ GCH ∧ (har‘𝐴) ∈ On) → 𝒫 𝐴 ≼ (𝒫 𝐴 +𝑐 (har‘𝐴)))
12	2, 1, 11	sylancl 694	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≼ (𝒫 𝐴 +𝑐 (har‘𝐴)))
13		ovex 6678	. . . . . . . 8 ⊢ (𝒫 𝐴 +𝑐 (har‘𝐴)) ∈ V
14	13	canth2 8113	. . . . . . 7 ⊢ (𝒫 𝐴 +𝑐 (har‘𝐴)) ≺ 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴))
15		pwcdaen 9007	. . . . . . . . 9 ⊢ ((𝒫 𝐴 ∈ GCH ∧ (har‘𝐴) ∈ On) → 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ≈ (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)))
16	2, 1, 15	sylancl 694	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ≈ (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)))
17		pwexg 4850	. . . . . . . . . . 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 1062	. . . . . . . . . . 11 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝐴 ∈ GCH)
20		harwdom 8495	. . . . . . . . . . 11 ⊢ (𝐴 ∈ GCH → (har‘𝐴) ≼* 𝒫 (𝐴 × 𝐴))
21		wdompwdom 8483	. . . . . . . . . . 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 8056	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)) ≼ (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)))
25		xpexg 6960	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ GCH ∧ 𝐴 ∈ GCH) → (𝐴 × 𝐴) ∈ V)
26	19, 19, 25	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 × 𝐴) ∈ V)
27		pwexg 4850	. . . . . . . . . . . . 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 9007	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 8007	. . . . . . . . . 10 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)))
32		enrefg 7987	. . . . . . . . . . . . . 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 9491	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 × 𝐴) ≈ 𝐴)
35	19, 8, 34	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 × 𝐴) ≈ 𝐴)
36		pwen 8133	. . . . . . . . . . . . . 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 8995	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 12 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ (𝒫 𝐴 +𝑐 𝒫 𝐴))
40		gchcdaidm 9490	. . . . . . . . . . . . 13 ⊢ ((𝒫 𝐴 ∈ GCH ∧ ¬ 𝒫 𝐴 ∈ Fin) → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴)
41	2, 10, 40	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝐴 +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴)
42		entr 8008	. . . . . . . . . . . 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 693	. . . . . . . . . . 11 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝐴 +𝑐 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝐴)
44		pwen 8133	. . . . . . . . . . 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 8008	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝒫 𝐴 × 𝒫 𝒫 (𝐴 × 𝐴)) ≈ 𝒫 𝒫 𝐴)
48		domentr 8015	. . . . . . . . 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 693	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝒫 𝐴 × 𝒫 (har‘𝐴)) ≼ 𝒫 𝒫 𝐴)
50		endomtr 8014	. . . . . . . 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 693	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝒫 𝐴 +𝑐 (har‘𝐴)) ≼ 𝒫 𝒫 𝐴)
52		sdomdomtr 8093	. . . . . . 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 695	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝒫 𝐴 +𝑐 (har‘𝐴)) ≺ 𝒫 𝒫 𝐴)
54		gchen1 9447	. . . . . 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 1327	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≈ (𝒫 𝐴 +𝑐 (har‘𝐴)))
56		cdacomen 9003	. . . . 5 ⊢ (𝒫 𝐴 +𝑐 (har‘𝐴)) ≈ ((har‘𝐴) +𝑐 𝒫 𝐴)
57		entr 8008	. . . . 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 694	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≈ ((har‘𝐴) +𝑐 𝒫 𝐴))
59	58	ensymd 8007	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ((har‘𝐴) +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴)
60		domentr 8015	. . 3 ⊢ (((har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝒫 𝐴) ∧ ((har‘𝐴) +𝑐 𝒫 𝐴) ≈ 𝒫 𝐴) → (har‘𝐴) ≼ 𝒫 𝐴)
61	4, 59, 60	syl2anc 693	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ 𝒫 𝐴)
62		gchcdaidm 9490	. . . . . 6 ⊢ ((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) → (𝐴 +𝑐 𝐴) ≈ 𝐴)
63	19, 8, 62	syl2anc 693	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 +𝑐 𝐴) ≈ 𝐴)
64		pwen 8133	. . . . 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 9010	. . . . . . . 8 ⊢ ((𝐴 ∈ GCH ∧ (har‘𝐴) ∈ On) → 𝐴 ≼ (𝐴 +𝑐 (har‘𝐴)))
67	19, 1, 66	sylancl 694	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝐴 ≼ (𝐴 +𝑐 (har‘𝐴)))
68		harndom 8469	. . . . . . . 8 ⊢ ¬ (har‘𝐴) ≼ 𝐴
69		cdadom3 9010	. . . . . . . . . . 11 ⊢ (((har‘𝐴) ∈ On ∧ 𝐴 ∈ GCH) → (har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝐴))
70	1, 19, 69	sylancr 695	. . . . . . . . . 10 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ ((har‘𝐴) +𝑐 𝐴))
71		cdacomen 9003	. . . . . . . . . 10 ⊢ ((har‘𝐴) +𝑐 𝐴) ≈ (𝐴 +𝑐 (har‘𝐴))
72		domentr 8015	. . . . . . . . . 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 694	. . . . . . . . 9 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≼ (𝐴 +𝑐 (har‘𝐴)))
74		domen2 8103	. . . . . . . . 9 ⊢ (𝐴 ≈ (𝐴 +𝑐 (har‘𝐴)) → ((har‘𝐴) ≼ 𝐴 ↔ (har‘𝐴) ≼ (𝐴 +𝑐 (har‘𝐴))))
75	73, 74	syl5ibrcom 237	. . . . . . . 8 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 ≈ (𝐴 +𝑐 (har‘𝐴)) → (har‘𝐴) ≼ 𝐴))
76	68, 75	mtoi 190	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → ¬ 𝐴 ≈ (𝐴 +𝑐 (har‘𝐴)))
77		brsdom 7978	. . . . . . 7 ⊢ (𝐴 ≺ (𝐴 +𝑐 (har‘𝐴)) ↔ (𝐴 ≼ (𝐴 +𝑐 (har‘𝐴)) ∧ ¬ 𝐴 ≈ (𝐴 +𝑐 (har‘𝐴))))
78	67, 76, 77	sylanbrc 698	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝐴 ≺ (𝐴 +𝑐 (har‘𝐴)))
79		canth2g 8114	. . . . . . . . 9 ⊢ (𝐴 ∈ GCH → 𝐴 ≺ 𝒫 𝐴)
80		sdomdom 7983	. . . . . . . . 9 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴)
81		cdadom1 9008	. . . . . . . . 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 9009	. . . . . . . . 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 8009	. . . . . . . 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 693	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 +𝑐 (har‘𝐴)) ≼ (𝒫 𝐴 +𝑐 𝒫 𝐴))
87		domentr 8015	. . . . . . 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 693	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 +𝑐 (har‘𝐴)) ≼ 𝒫 𝐴)
89		gchen2 9448	. . . . . 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 1327	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (𝐴 +𝑐 (har‘𝐴)) ≈ 𝒫 𝐴)
91	90	ensymd 8007	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 𝐴 ≈ (𝐴 +𝑐 (har‘𝐴)))
92		entr 8008	. . . 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 693	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → 𝒫 (𝐴 +𝑐 𝐴) ≈ (𝐴 +𝑐 (har‘𝐴)))
94		endom 7982	. . 3 ⊢ (𝒫 (𝐴 +𝑐 𝐴) ≈ (𝐴 +𝑐 (har‘𝐴)) → 𝒫 (𝐴 +𝑐 𝐴) ≼ (𝐴 +𝑐 (har‘𝐴)))
95		pwcdadom 9038	. . 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 8080	. 2 ⊢ (((har‘𝐴) ≼ 𝒫 𝐴 ∧ 𝒫 𝐴 ≼ (har‘𝐴)) → (har‘𝐴) ≈ 𝒫 𝐴)
98	61, 96, 97	syl2anc 693	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐴 ∈ GCH ∧ 𝒫 𝐴 ∈ GCH) → (har‘𝐴) ≈ 𝒫 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1037   ∈ wcel 1990  Vcvv 3200  𝒫 cpw 4158   class class class wbr 4653   × cxp 5112  Oncon0 5723  ‘cfv 5888  (class class class)co 6650  ωcom 7065   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  harchar 8461   ≼^* cwdom 8462   +_𝑐 ccda 8989  GCHcgch 9442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-har 8463  df-wdom 8464  df-cnf 8559  df-card 8765  df-cda 8990  df-fin4 9109  df-gch 9443

This theorem is referenced by:  gchacg  9502