Theorem cfpwsdom 9852

Description: A corollary of Konig's Theorem konigth 9837. 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 7048	. . . . . . . . 9 ⊢ (𝐵 ↑𝑚 (ℵ‘𝐴)) ∈ V
2	1	cardid 9815	. . . . . . . 8 ⊢ (card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) ≈ (𝐵 ↑𝑚 (ℵ‘𝐴))
3	2	ensymi 8407	. . . . . . 7 ⊢ (𝐵 ↑𝑚 (ℵ‘𝐴)) ≈ (card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))
4		fvex 6551	. . . . . . . . . . . . . 14 ⊢ (ℵ‘𝐴) ∈ V
5	4	canth2 8517	. . . . . . . . . . . . 13 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
6	4	pw2en 8471	. . . . . . . . . . . . 13 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2o ↑𝑚 (ℵ‘𝐴))
7		sdomentr 8498	. . . . . . . . . . . . 13 ⊢ (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2o ↑𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2o ↑𝑚 (ℵ‘𝐴)))
8	5, 6, 7	mp2an 688	. . . . . . . . . . . 12 ⊢ (ℵ‘𝐴) ≺ (2o ↑𝑚 (ℵ‘𝐴))
9		mapdom1 8529	. . . . . . . . . . . 12 ⊢ (2o ≼ 𝐵 → (2o ↑𝑚 (ℵ‘𝐴)) ≼ (𝐵 ↑𝑚 (ℵ‘𝐴)))
10		sdomdomtr 8497	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (𝐵 ↑𝑚 (ℵ‘𝐴)))
12		ficard 9833	. . . . . . . . . . . . . . . . 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 8962	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑𝑚 (ℵ‘𝐴)) ∈ Fin → (𝐵 ↑𝑚 (ℵ‘𝐴)) ≼ ω)
15	13, 14	sylbir 236	. . . . . . . . . . . . . . 15 ⊢ ((card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) ∈ ω → (𝐵 ↑𝑚 (ℵ‘𝐴)) ≼ ω)
16		alephgeom 9354	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		alephon 9341	. . . . . . . . . . . . . . . . 17 ⊢ (ℵ‘𝐴) ∈ On
18		ssdomg 8403	. . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
21		domtr 8410	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ↑𝑚 (ℵ‘𝐴)) ≼ ω ∧ ω ≼ (ℵ‘𝐴)) → (𝐵 ↑𝑚 (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
22	15, 20, 21	syl2an 595	. . . . . . . . . . . . . 14 ⊢ (((card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → (𝐵 ↑𝑚 (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
23		domnsym 8490	. . . . . . . . . . . . . 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 414	. . . . . . . . . . . 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 9234	. . . . . . . . . . . 12 ⊢ (card‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) = (card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))
28		iscard3 9365	. . . . . . . . . . . . 13 ⊢ ((card‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) = (card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) ↔ (card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
29		elun 4046	. . . . . . . . . . . . 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 843	. . . . . . . . . . . . 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 298	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 9337	. . . . . . . . . . 11 ⊢ ℵ Fn On
35		fvelrnb 6594	. . . . . . . . . . 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 252	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))))
38		eqid 2795	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦)))
39	38	pwcfsdom 9851	. . . . . . . . . . 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 6538	. . . . . . . . . . . . 13 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))))
42	40, 41	oveq12d 7034	. . . . . . . . . . . 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 4975	. . . . . . . . . . 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 234	. . . . . . . . . 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 3245	. . . . . . . . 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 407	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) ≺ ((card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴))))))
48		ensdomtr 8500	. . . . . . 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 587	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (𝐵 ↑𝑚 (ℵ‘𝐴)) ≺ ((card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) ↑𝑚 (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴))))))
50		fvex 6551	. . . . . . . . 9 ⊢ (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) ∈ V
51	50	enref 8390	. . . . . . . 8 ⊢ (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴))))
52		mapen 8528	. . . . . . . 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 688	. . . . . . 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 8530	. . . . . . . 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 1453	. . . . . . 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 8411	. . . . . 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 8498	. . . . . 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 586	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (𝐵 ↑𝑚 (ℵ‘𝐴)) ≺ (𝐵 ↑𝑚 ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))))))
60	4	xpdom2 8459	. . . . . . . . . 10 ⊢ ((cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
61	16	biimpi 217	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
62		infxpen 9286	. . . . . . . . . . 11 ⊢ (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
63	17, 61, 62	sylancr 587	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
64		domentr 8416	. . . . . . . . . 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 595	. . . . . . . . 9 ⊢ (((cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
66		nsuceq0 6146	. . . . . . . . . . 11 ⊢ suc 1o ≠ ∅
67		dom0 8492	. . . . . . . . . . 11 ⊢ (suc 1o ≼ ∅ ↔ suc 1o = ∅)
68	66, 67	nemtbir 3080	. . . . . . . . . 10 ⊢ ¬ suc 1o ≼ ∅
69		df-2o 7954	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
70	69	breq1i 4969	. . . . . . . . . . . . 13 ⊢ (2o ≼ 𝐵 ↔ suc 1o ≼ 𝐵)
71		breq2 4966	. . . . . . . . . . . . 13 ⊢ (𝐵 = ∅ → (suc 1o ≼ 𝐵 ↔ suc 1o ≼ ∅))
72	70, 71	syl5bb 284	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (2o ≼ 𝐵 ↔ suc 1o ≼ ∅))
73	72	biimpcd 250	. . . . . . . . . . 11 ⊢ (2o ≼ 𝐵 → (𝐵 = ∅ → suc 1o ≼ ∅))
74	73	adantld 491	. . . . . . . . . 10 ⊢ (2o ≼ 𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1o ≼ ∅))
75	68, 74	mtoi 200	. . . . . . . . 9 ⊢ (2o ≼ 𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
76		mapdom2 8535	. . . . . . . . 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 595	. . . . . . . 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 8490	. . . . . . . 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 458	. . . . . 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 9824	. . . . . 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 688	. . . . 5 ⊢ ((cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))))
85	84	biimpri 229	. . . 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 413	. 2 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴))))))
88		fndm 6325	. . . . . 6 ⊢ (ℵ Fn On → dom ℵ = On)
89	34, 88	ax-mp 5	. . . . 5 ⊢ dom ℵ = On
90	89	eleq2i 2874	. . . 4 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
91		ndmfv 6568	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
92	90, 91	sylnbir 332	. . 3 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
93		1n0 7970	. . . . . 6 ⊢ 1o ≠ ∅
94		1oex 7961	. . . . . . 7 ⊢ 1o ∈ V
95	94	0sdom 8495	. . . . . 6 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
96	93, 95	mpbir 232	. . . . 5 ⊢ ∅ ≺ 1o
97		id 22	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
98		oveq2 7024	. . . . . . . . . . 11 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑𝑚 (ℵ‘𝐴)) = (𝐵 ↑𝑚 ∅))
99		map0e 8296	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → (𝐵 ↑𝑚 ∅) = 1o)
100	54, 99	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐵 ↑𝑚 ∅) = 1o
101	98, 100	syl6eq 2847	. . . . . . . . . 10 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑𝑚 (ℵ‘𝐴)) = 1o)
102	101	fveq2d 6542	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) = (card‘1o))
103		1onn 8115	. . . . . . . . . 10 ⊢ 1o ∈ ω
104		cardnn 9238	. . . . . . . . . 10 ⊢ (1o ∈ ω → (card‘1o) = 1o)
105	103, 104	ax-mp 5	. . . . . . . . 9 ⊢ (card‘1o) = 1o
106	102, 105	syl6eq 2847	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑𝑚 (ℵ‘𝐴))) = 1o)
107	106	fveq2d 6542	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) = (cf‘1o))
108		df-1o 7953	. . . . . . . . 9 ⊢ 1o = suc ∅
109	108	fveq2i 6541	. . . . . . . 8 ⊢ (cf‘1o) = (cf‘suc ∅)
110		0elon 6119	. . . . . . . . 9 ⊢ ∅ ∈ On
111		cfsuc 9525	. . . . . . . . 9 ⊢ (∅ ∈ On → (cf‘suc ∅) = 1o)
112	110, 111	ax-mp 5	. . . . . . . 8 ⊢ (cf‘suc ∅) = 1o
113	109, 112	eqtri 2819	. . . . . . 7 ⊢ (cf‘1o) = 1o
114	107, 113	syl6eq 2847	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) = 1o)
115	97, 114	breq12d 4975	. . . . 5 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))) ↔ ∅ ≺ 1o))
116	96, 115	mpbiri 259	. . . 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 183	1 ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑𝑚 (ℵ‘𝐴)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   = wceq 1522   ∈ wcel 2081   ≠ wne 2984  ∃wrex 3106  Vcvv 3437   ∪ cun 3857   ⊆ wss 3859  ∅c0 4211  𝒫 cpw 4453   class class class wbr 4962   ↦ cmpt 5041   × cxp 5441  dom cdm 5443  ran crn 5444  Oncon0 6066  suc csuc 6068   Fn wfn 6220  ‘cfv 6225  (class class class)co 7016  ωcom 7436  1_oc1o 7946  2_oc2o 7947   ↑_𝑚 cmap 8256   ≈ cen 8354   ≼ cdom 8355   ≺ csdm 8356  Fincfn 8357  harchar 8866  cardccrd 9210  ℵcale 9211  cfccf 9212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-ac2 9731

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-smo 7835  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-oi 8820  df-har 8868  df-card 9214  df-aleph 9215  df-cf 9216  df-acn 9217  df-ac 9388

This theorem is referenced by:  alephom  9853