Theorem cfpwsdom 10000

Description: A corollary of Konig's Theorem konigth 9985. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)

Hypothesis

Ref	Expression
cfpwsdom.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
cfpwsdom	⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))

Proof of Theorem cfpwsdom

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7183	. . . . . . . . 9 ⊢ (𝐵 ↑m (ℵ‘𝐴)) ∈ V
2	1	cardid 9963	. . . . . . . 8 ⊢ (card‘(𝐵 ↑m (ℵ‘𝐴))) ≈ (𝐵 ↑m (ℵ‘𝐴))
3	2	ensymi 8553	. . . . . . 7 ⊢ (𝐵 ↑m (ℵ‘𝐴)) ≈ (card‘(𝐵 ↑m (ℵ‘𝐴)))
4		fvex 6678	. . . . . . . . . . . . . 14 ⊢ (ℵ‘𝐴) ∈ V
5	4	canth2 8664	. . . . . . . . . . . . 13 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
6	4	pw2en 8618	. . . . . . . . . . . . 13 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2o ↑m (ℵ‘𝐴))
7		sdomentr 8645	. . . . . . . . . . . . 13 ⊢ (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2o ↑m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2o ↑m (ℵ‘𝐴)))
8	5, 6, 7	mp2an 690	. . . . . . . . . . . 12 ⊢ (ℵ‘𝐴) ≺ (2o ↑m (ℵ‘𝐴))
9		mapdom1 8676	. . . . . . . . . . . 12 ⊢ (2o ≼ 𝐵 → (2o ↑m (ℵ‘𝐴)) ≼ (𝐵 ↑m (ℵ‘𝐴)))
10		sdomdomtr 8644	. . . . . . . . . . . 12 ⊢ (((ℵ‘𝐴) ≺ (2o ↑m (ℵ‘𝐴)) ∧ (2o ↑m (ℵ‘𝐴)) ≼ (𝐵 ↑m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)))
11	8, 9, 10	sylancr 589	. . . . . . . . . . 11 ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)))
12		ficard 9981	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ↑m (ℵ‘𝐴)) ∈ V → ((𝐵 ↑m (ℵ‘𝐴)) ∈ Fin ↔ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω))
13	1, 12	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑m (ℵ‘𝐴)) ∈ Fin ↔ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω)
14		fict 9110	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑m (ℵ‘𝐴)) ∈ Fin → (𝐵 ↑m (ℵ‘𝐴)) ≼ ω)
15	13, 14	sylbir 237	. . . . . . . . . . . . . . 15 ⊢ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → (𝐵 ↑m (ℵ‘𝐴)) ≼ ω)
16		alephgeom 9502	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		alephon 9489	. . . . . . . . . . . . . . . . 17 ⊢ (ℵ‘𝐴) ∈ On
18		ssdomg 8549	. . . . . . . . . . . . . . . . 17 ⊢ ((ℵ‘𝐴) ∈ On → (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴)))
19	17, 18	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (ω ⊆ (ℵ‘𝐴) → ω ≼ (ℵ‘𝐴))
20	16, 19	sylbi 219	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
21		domtr 8556	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ↑m (ℵ‘𝐴)) ≼ ω ∧ ω ≼ (ℵ‘𝐴)) → (𝐵 ↑m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
22	15, 20, 21	syl2an 597	. . . . . . . . . . . . . 14 ⊢ (((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → (𝐵 ↑m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
23		domnsym 8637	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ↑m (ℵ‘𝐴)) ≼ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)))
24	22, 23	syl 17	. . . . . . . . . . . . 13 ⊢ (((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → ¬ (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)))
25	24	expcom 416	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → ¬ (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴))))
26	25	con2d 136	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)) → ¬ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω))
27		cardidm 9382	. . . . . . . . . . . 12 ⊢ (card‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (card‘(𝐵 ↑m (ℵ‘𝐴)))
28		iscard3 9513	. . . . . . . . . . . . 13 ⊢ ((card‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (card‘(𝐵 ↑m (ℵ‘𝐴))) ↔ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
29		elun 4125	. . . . . . . . . . . . 13 ⊢ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ) ↔ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ))
30		df-or 844	. . . . . . . . . . . . 13 ⊢ (((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω ∨ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ) ↔ (¬ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ))
31	28, 29, 30	3bitri 299	. . . . . . . . . . . 12 ⊢ ((card‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (card‘(𝐵 ↑m (ℵ‘𝐴))) ↔ (¬ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ))
32	27, 31	mpbi 232	. . . . . . . . . . 11 ⊢ (¬ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ)
33	11, 26, 32	syl56 36	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ))
34		alephfnon 9485	. . . . . . . . . . 11 ⊢ ℵ Fn On
35		fvelrnb 6721	. . . . . . . . . . 11 ⊢ (ℵ Fn On → ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴)))))
36	34, 35	ax-mp 5	. . . . . . . . . 10 ⊢ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ ↔ ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))))
37	33, 36	syl6ib 253	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴)))))
38		eqid 2821	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦)))
39	38	pwcfsdom 9999	. . . . . . . . . . 11 ⊢ (ℵ‘𝑥) ≺ ((ℵ‘𝑥) ↑m (cf‘(ℵ‘𝑥)))
40		id 22	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → (ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))))
41		fveq2 6665	. . . . . . . . . . . . 13 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
42	40, 41	oveq12d 7168	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → ((ℵ‘𝑥) ↑m (cf‘(ℵ‘𝑥))) = ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
43	40, 42	breq12d 5072	. . . . . . . . . . 11 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → ((ℵ‘𝑥) ≺ ((ℵ‘𝑥) ↑m (cf‘(ℵ‘𝑥))) ↔ (card‘(𝐵 ↑m (ℵ‘𝐴))) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))))
44	39, 43	mpbii 235	. . . . . . . . . 10 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → (card‘(𝐵 ↑m (ℵ‘𝐴))) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
45	44	rexlimivw 3282	. . . . . . . . 9 ⊢ (∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → (card‘(𝐵 ↑m (ℵ‘𝐴))) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
46	37, 45	syl6 35	. . . . . . . 8 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → (card‘(𝐵 ↑m (ℵ‘𝐴))) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))))
47	46	imp 409	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (card‘(𝐵 ↑m (ℵ‘𝐴))) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
48		ensdomtr 8647	. . . . . . 7 ⊢ (((𝐵 ↑m (ℵ‘𝐴)) ≈ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∧ (card‘(𝐵 ↑m (ℵ‘𝐴))) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))) → (𝐵 ↑m (ℵ‘𝐴)) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
49	3, 47, 48	sylancr 589	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (𝐵 ↑m (ℵ‘𝐴)) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
50		fvex 6678	. . . . . . . . 9 ⊢ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ∈ V
51	50	enref 8536	. . . . . . . 8 ⊢ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))
52		mapen 8675	. . . . . . . 8 ⊢ (((card‘(𝐵 ↑m (ℵ‘𝐴))) ≈ (𝐵 ↑m (ℵ‘𝐴)) ∧ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) → ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≈ ((𝐵 ↑m (ℵ‘𝐴)) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
53	2, 51, 52	mp2an 690	. . . . . . 7 ⊢ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≈ ((𝐵 ↑m (ℵ‘𝐴)) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
54		cfpwsdom.1	. . . . . . . 8 ⊢ 𝐵 ∈ V
55		mapxpen 8677	. . . . . . . 8 ⊢ ((𝐵 ∈ V ∧ (ℵ‘𝐴) ∈ On ∧ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ∈ V) → ((𝐵 ↑m (ℵ‘𝐴)) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≈ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))))
56	54, 17, 50, 55	mp3an 1457	. . . . . . 7 ⊢ ((𝐵 ↑m (ℵ‘𝐴)) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≈ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
57	53, 56	entri 8557	. . . . . 6 ⊢ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≈ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
58		sdomentr 8645	. . . . . 6 ⊢ (((𝐵 ↑m (ℵ‘𝐴)) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ∧ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≈ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))) → (𝐵 ↑m (ℵ‘𝐴)) ≺ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))))
59	49, 57, 58	sylancl 588	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (𝐵 ↑m (ℵ‘𝐴)) ≺ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))))
60	4	xpdom2 8606	. . . . . . . . . 10 ⊢ ((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
61	16	biimpi 218	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
62		infxpen 9434	. . . . . . . . . . 11 ⊢ (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
63	17, 61, 62	sylancr 589	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
64		domentr 8562	. . . . . . . . . 10 ⊢ ((((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)) ∧ ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
65	60, 63, 64	syl2an 597	. . . . . . . . 9 ⊢ (((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
66		nsuceq0 6266	. . . . . . . . . . 11 ⊢ suc 1o ≠ ∅
67		dom0 8639	. . . . . . . . . . 11 ⊢ (suc 1o ≼ ∅ ↔ suc 1o = ∅)
68	66, 67	nemtbir 3112	. . . . . . . . . 10 ⊢ ¬ suc 1o ≼ ∅
69		df-2o 8097	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
70	69	breq1i 5066	. . . . . . . . . . . . 13 ⊢ (2o ≼ 𝐵 ↔ suc 1o ≼ 𝐵)
71		breq2 5063	. . . . . . . . . . . . 13 ⊢ (𝐵 = ∅ → (suc 1o ≼ 𝐵 ↔ suc 1o ≼ ∅))
72	70, 71	syl5bb 285	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (2o ≼ 𝐵 ↔ suc 1o ≼ ∅))
73	72	biimpcd 251	. . . . . . . . . . 11 ⊢ (2o ≼ 𝐵 → (𝐵 = ∅ → suc 1o ≼ ∅))
74	73	adantld 493	. . . . . . . . . 10 ⊢ (2o ≼ 𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1o ≼ ∅))
75	68, 74	mtoi 201	. . . . . . . . 9 ⊢ (2o ≼ 𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
76		mapdom2 8682	. . . . . . . . 9 ⊢ ((((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≼ (ℵ‘𝐴) ∧ ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅)) → (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))) ≼ (𝐵 ↑m (ℵ‘𝐴)))
77	65, 75, 76	syl2an 597	. . . . . . . 8 ⊢ ((((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2o ≼ 𝐵) → (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))) ≼ (𝐵 ↑m (ℵ‘𝐴)))
78		domnsym 8637	. . . . . . . 8 ⊢ ((𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))) ≼ (𝐵 ↑m (ℵ‘𝐴)) → ¬ (𝐵 ↑m (ℵ‘𝐴)) ≺ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))))
79	77, 78	syl 17	. . . . . . 7 ⊢ ((((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) ∧ 2o ≼ 𝐵) → ¬ (𝐵 ↑m (ℵ‘𝐴)) ≺ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))))
80	79	expl 460	. . . . . 6 ⊢ ((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → ¬ (𝐵 ↑m (ℵ‘𝐴)) ≺ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))))
81	80	com12 32	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → ((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ¬ (𝐵 ↑m (ℵ‘𝐴)) ≺ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))))
82	59, 81	mt2d 138	. . . 4 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → ¬ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
83		domtri 9972	. . . . . 6 ⊢ (((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ∈ V ∧ (ℵ‘𝐴) ∈ V) → ((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
84	50, 4, 83	mp2an 690	. . . . 5 ⊢ ((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
85	84	biimpri 230	. . . 4 ⊢ (¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
86	82, 85	nsyl2 143	. . 3 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
87	86	ex 415	. 2 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
88		fndm 6450	. . . . . 6 ⊢ (ℵ Fn On → dom ℵ = On)
89	34, 88	ax-mp 5	. . . . 5 ⊢ dom ℵ = On
90	89	eleq2i 2904	. . . 4 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
91		ndmfv 6695	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
92	90, 91	sylnbir 333	. . 3 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
93		1n0 8113	. . . . . 6 ⊢ 1o ≠ ∅
94		1oex 8104	. . . . . . 7 ⊢ 1o ∈ V
95	94	0sdom 8642	. . . . . 6 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
96	93, 95	mpbir 233	. . . . 5 ⊢ ∅ ≺ 1o
97		id 22	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
98		oveq2 7158	. . . . . . . . . . 11 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑m (ℵ‘𝐴)) = (𝐵 ↑m ∅))
99		map0e 8440	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → (𝐵 ↑m ∅) = 1o)
100	54, 99	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐵 ↑m ∅) = 1o
101	98, 100	syl6eq 2872	. . . . . . . . . 10 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑m (ℵ‘𝐴)) = 1o)
102	101	fveq2d 6669	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑m (ℵ‘𝐴))) = (card‘1o))
103		1onn 8259	. . . . . . . . . 10 ⊢ 1o ∈ ω
104		cardnn 9386	. . . . . . . . . 10 ⊢ (1o ∈ ω → (card‘1o) = 1o)
105	103, 104	ax-mp 5	. . . . . . . . 9 ⊢ (card‘1o) = 1o
106	102, 105	syl6eq 2872	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑m (ℵ‘𝐴))) = 1o)
107	106	fveq2d 6669	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (cf‘1o))
108		df-1o 8096	. . . . . . . . 9 ⊢ 1o = suc ∅
109	108	fveq2i 6668	. . . . . . . 8 ⊢ (cf‘1o) = (cf‘suc ∅)
110		0elon 6239	. . . . . . . . 9 ⊢ ∅ ∈ On
111		cfsuc 9673	. . . . . . . . 9 ⊢ (∅ ∈ On → (cf‘suc ∅) = 1o)
112	110, 111	ax-mp 5	. . . . . . . 8 ⊢ (cf‘suc ∅) = 1o
113	109, 112	eqtri 2844	. . . . . . 7 ⊢ (cf‘1o) = 1o
114	107, 113	syl6eq 2872	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = 1o)
115	97, 114	breq12d 5072	. . . . 5 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ↔ ∅ ≺ 1o))
116	96, 115	mpbiri 260	. . . 4 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
117	116	a1d 25	. . 3 ⊢ ((ℵ‘𝐴) = ∅ → (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
118	92, 117	syl 17	. 2 ⊢ (¬ 𝐴 ∈ On → (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
119	87, 118	pm2.61i 184	1 ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∃wrex 3139  Vcvv 3495   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539   class class class wbr 5059   ↦ cmpt 5139   × cxp 5548  dom cdm 5550  ran crn 5551  Oncon0 6186  suc csuc 6188   Fn wfn 6345  ‘cfv 6350  (class class class)co 7150  ωcom 7574  1_oc1o 8089  2_oc2o 8090   ↑_m cmap 8400   ≈ cen 8500   ≼ cdom 8501   ≺ csdm 8502  Fincfn 8503  harchar 9014  cardccrd 9358  ℵcale 9359  cfccf 9360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-inf2 9098  ax-ac2 9879

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-smo 7977  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-oi 8968  df-har 9016  df-card 9362  df-aleph 9363  df-cf 9364  df-acn 9365  df-ac 9536

This theorem is referenced by:  alephom  10001