Theorem cfpwsdom 10596

Description: A corollary of Konig's Theorem konigth 10581. 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 7447	. . . . . . . . 9 ⊢ (𝐵 ↑m (ℵ‘𝐴)) ∈ V
2	1	cardid 10558	. . . . . . . 8 ⊢ (card‘(𝐵 ↑m (ℵ‘𝐴))) ≈ (𝐵 ↑m (ℵ‘𝐴))
3	2	ensymi 9013	. . . . . . 7 ⊢ (𝐵 ↑m (ℵ‘𝐴)) ≈ (card‘(𝐵 ↑m (ℵ‘𝐴)))
4		fvex 6892	. . . . . . . . . . . . . 14 ⊢ (ℵ‘𝐴) ∈ V
5	4	canth2 9131	. . . . . . . . . . . . 13 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
6	4	pw2en 9085	. . . . . . . . . . . . 13 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2o ↑m (ℵ‘𝐴))
7		sdomentr 9112	. . . . . . . . . . . . 13 ⊢ (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2o ↑m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2o ↑m (ℵ‘𝐴)))
8	5, 6, 7	mp2an 705	. . . . . . . . . . . 12 ⊢ (ℵ‘𝐴) ≺ (2o ↑m (ℵ‘𝐴))
9		mapdom1 9143	. . . . . . . . . . . 12 ⊢ (2o ≼ 𝐵 → (2o ↑m (ℵ‘𝐴)) ≼ (𝐵 ↑m (ℵ‘𝐴)))
10		sdomdomtr 9111	. . . . . . . . . . . 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 599	. . . . . . . . . . 11 ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)))
12		ficard 10576	. . . . . . . . . . . . . . . . 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 9635	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑m (ℵ‘𝐴)) ∈ Fin → (𝐵 ↑m (ℵ‘𝐴)) ≼ ω)
15	13, 14	sylbir 238	. . . . . . . . . . . . . . 15 ⊢ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → (𝐵 ↑m (ℵ‘𝐴)) ≼ ω)
16		alephgeom 10088	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		alephon 10075	. . . . . . . . . . . . . . . . 17 ⊢ (ℵ‘𝐴) ∈ On
18		ssdomg 9009	. . . . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
21		domtr 9016	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ↑m (ℵ‘𝐴)) ≼ ω ∧ ω ≼ (ℵ‘𝐴)) → (𝐵 ↑m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
22	15, 20, 21	syl2an 608	. . . . . . . . . . . . . 14 ⊢ (((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → (𝐵 ↑m (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
23		domnsym 9104	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ↑m (ℵ‘𝐴)) ≼ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)))
24	22, 23	syl 18	. . . . . . . . . . . . 13 ⊢ (((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω ∧ 𝐴 ∈ On) → ¬ (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)))
25	24	expcom 419	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → ¬ (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴))))
26	25	con2d 135	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)) → ¬ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω))
27		cardidm 9967	. . . . . . . . . . . 12 ⊢ (card‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (card‘(𝐵 ↑m (ℵ‘𝐴)))
28		iscard3 10099	. . . . . . . . . . . . 13 ⊢ ((card‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (card‘(𝐵 ↑m (ℵ‘𝐴))) ↔ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
29		elun 4100	. . . . . . . . . . . . 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 862	. . . . . . . . . . . . 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 300	. . . . . . . . . . . 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 233	. . . . . . . . . . 11 ⊢ (¬ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ)
33	11, 26, 32	syl56 37	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ran ℵ))
34		alephfnon 10071	. . . . . . . . . . 11 ⊢ ℵ Fn On
35		fvelrnb 6939	. . . . . . . . . . 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	imbitrdi 254	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴)))))
38		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦)))
39	38	pwcfsdom 10595	. . . . . . . . . . 11 ⊢ (ℵ‘𝑥) ≺ ((ℵ‘𝑥) ↑m (cf‘(ℵ‘𝑥)))
40		id 23	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → (ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))))
41		fveq2 6879	. . . . . . . . . . . . 13 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
42	40, 41	oveq12d 7432	. . . . . . . . . . . 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 5116	. . . . . . . . . . 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 236	. . . . . . . . . 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 3159	. . . . . . . . 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 36	. . . . . . . 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 412	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (card‘(𝐵 ↑m (ℵ‘𝐴))) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
48		ensdomtr 9114	. . . . . . 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 599	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (𝐵 ↑m (ℵ‘𝐴)) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
50		fvex 6892	. . . . . . . . 9 ⊢ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ∈ V
51	50	enref 8994	. . . . . . . 8 ⊢ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))
52		mapen 9142	. . . . . . . 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 705	. . . . . . 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 9144	. . . . . . . 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 1490	. . . . . . 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 9017	. . . . . 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 9112	. . . . . 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 598	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (𝐵 ↑m (ℵ‘𝐴)) ≺ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))))
60	4	xpdom2 9073	. . . . . . . . . 10 ⊢ ((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
61	16	biimpi 219	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
62		infxpen 10020	. . . . . . . . . . 11 ⊢ (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
63	17, 61, 62	sylancr 599	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
64		domentr 9022	. . . . . . . . . 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 608	. . . . . . . . 9 ⊢ (((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ∧ 𝐴 ∈ On) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≼ (ℵ‘𝐴))
66		nsuceq0 6443	. . . . . . . . . . 11 ⊢ suc 1o ≠ ∅
67		dom0 9106	. . . . . . . . . . 11 ⊢ (suc 1o ≼ ∅ ↔ suc 1o = ∅)
68	66, 67	nemtbir 3051	. . . . . . . . . 10 ⊢ ¬ suc 1o ≼ ∅
69		df-2o 8459	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
70	69	breq1i 5110	. . . . . . . . . . . . 13 ⊢ (2o ≼ 𝐵 ↔ suc 1o ≼ 𝐵)
71		breq2 5107	. . . . . . . . . . . . 13 ⊢ (𝐵 = ∅ → (suc 1o ≼ 𝐵 ↔ suc 1o ≼ ∅))
72	70, 71	bitrid 286	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (2o ≼ 𝐵 ↔ suc 1o ≼ ∅))
73	72	biimpcd 252	. . . . . . . . . . 11 ⊢ (2o ≼ 𝐵 → (𝐵 = ∅ → suc 1o ≼ ∅))
74	73	adantld 496	. . . . . . . . . 10 ⊢ (2o ≼ 𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1o ≼ ∅))
75	68, 74	mtoi 202	. . . . . . . . 9 ⊢ (2o ≼ 𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
76		mapdom2 9149	. . . . . . . . 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 608	. . . . . . . 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 9104	. . . . . . . 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 18	. . . . . . 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 463	. . . . . 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 33	. . . . 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 137	. . . 4 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → ¬ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
83		domtri 10567	. . . . . 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 705	. . . . 5 ⊢ ((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
85	84	biimpri 231	. . . 4 ⊢ (¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
86	82, 85	nsyl2 142	. . 3 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
87	86	ex 418	. 2 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
88		fndm 6636	. . . . . 6 ⊢ (ℵ Fn On → dom ℵ = On)
89	34, 88	ax-mp 5	. . . . 5 ⊢ dom ℵ = On
90	89	eleq2i 2852	. . . 4 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
91		ndmfv 6911	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
92	90, 91	sylnbir 334	. . 3 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
93		1n0 8477	. . . . . 6 ⊢ 1o ≠ ∅
94		1oex 8468	. . . . . . 7 ⊢ 1o ∈ V
95	94	0sdom 9109	. . . . . 6 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
96	93, 95	mpbir 234	. . . . 5 ⊢ ∅ ≺ 1o
97		id 23	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
98		oveq2 7422	. . . . . . . . . . 11 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑m (ℵ‘𝐴)) = (𝐵 ↑m ∅))
99		map0e 8892	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → (𝐵 ↑m ∅) = 1o)
100	54, 99	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐵 ↑m ∅) = 1o
101	98, 100	eqtrdi 2811	. . . . . . . . . 10 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑m (ℵ‘𝐴)) = 1o)
102	101	fveq2d 6883	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑m (ℵ‘𝐴))) = (card‘1o))
103		1onn 8631	. . . . . . . . . 10 ⊢ 1o ∈ ω
104		cardnn 9971	. . . . . . . . . 10 ⊢ (1o ∈ ω → (card‘1o) = 1o)
105	103, 104	ax-mp 5	. . . . . . . . 9 ⊢ (card‘1o) = 1o
106	102, 105	eqtrdi 2811	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑m (ℵ‘𝐴))) = 1o)
107	106	fveq2d 6883	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (cf‘1o))
108		df-1o 8458	. . . . . . . . 9 ⊢ 1o = suc ∅
109	108	fveq2i 6882	. . . . . . . 8 ⊢ (cf‘1o) = (cf‘suc ∅)
110		0elon 6413	. . . . . . . . 9 ⊢ ∅ ∈ On
111		cfsuc 10262	. . . . . . . . 9 ⊢ (∅ ∈ On → (cf‘suc ∅) = 1o)
112	110, 111	ax-mp 5	. . . . . . . 8 ⊢ (cf‘suc ∅) = 1o
113	109, 112	eqtri 2783	. . . . . . 7 ⊢ (cf‘1o) = 1o
114	107, 113	eqtrdi 2811	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = 1o)
115	97, 114	breq12d 5116	. . . . 5 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ↔ ∅ ≺ 1o))
116	96, 115	mpbiri 261	. . . 4 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
117	116	a1d 26	. . 3 ⊢ ((ℵ‘𝐴) = ∅ → (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
118	92, 117	syl 18	. 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)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086  Vcvv 3450   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186   × cxp 5653  dom cdm 5655  ran crn 5656  Oncon0 6357  suc csuc 6359   Fn wfn 6528  ‘cfv 6533  (class class class)co 7414  ωcom 7863  1_oc1o 8451  2_oc2o 8452   ↑_m cmap 8829   ≈ cen 8952   ≼ cdom 8953   ≺ csdm 8954  Fincfn 8955  harchar 9531  cardccrd 9943  ℵcale 9944  cfccf 9945

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-inf2 9623  ax-ac2 10468

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-smo 8336  df-recs 8361  df-rdg 8400  df-1o 8458  df-2o 8459  df-er 8699  df-map 8831  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9485  df-har 9532  df-card 9947  df-aleph 9948  df-cf 9949  df-acn 9950  df-ac 10122

This theorem is used by:  alephom  10597