Theorem cfpwsdom 10507

Description: A corollary of Konig's Theorem konigth 10492. 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 7400	. . . . . . . . 9 ⊢ (𝐵 ↑m (ℵ‘𝐴)) ∈ V
2	1	cardid 10469	. . . . . . . 8 ⊢ (card‘(𝐵 ↑m (ℵ‘𝐴))) ≈ (𝐵 ↑m (ℵ‘𝐴))
3	2	ensymi 8951	. . . . . . 7 ⊢ (𝐵 ↑m (ℵ‘𝐴)) ≈ (card‘(𝐵 ↑m (ℵ‘𝐴)))
4		fvex 6853	. . . . . . . . . . . . . 14 ⊢ (ℵ‘𝐴) ∈ V
5	4	canth2 9068	. . . . . . . . . . . . 13 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
6	4	pw2en 9022	. . . . . . . . . . . . 13 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2o ↑m (ℵ‘𝐴))
7		sdomentr 9049	. . . . . . . . . . . . 13 ⊢ (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2o ↑m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2o ↑m (ℵ‘𝐴)))
8	5, 6, 7	mp2an 693	. . . . . . . . . . . 12 ⊢ (ℵ‘𝐴) ≺ (2o ↑m (ℵ‘𝐴))
9		mapdom1 9080	. . . . . . . . . . . 12 ⊢ (2o ≼ 𝐵 → (2o ↑m (ℵ‘𝐴)) ≼ (𝐵 ↑m (ℵ‘𝐴)))
10		sdomdomtr 9048	. . . . . . . . . . . 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 588	. . . . . . . . . . 11 ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)))
12		ficard 10487	. . . . . . . . . . . . . . . . 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 9574	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ↑m (ℵ‘𝐴)) ∈ Fin → (𝐵 ↑m (ℵ‘𝐴)) ≼ ω)
15	13, 14	sylbir 235	. . . . . . . . . . . . . . 15 ⊢ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → (𝐵 ↑m (ℵ‘𝐴)) ≼ ω)
16		alephgeom 10004	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		alephon 9991	. . . . . . . . . . . . . . . . 17 ⊢ (ℵ‘𝐴) ∈ On
18		ssdomg 8947	. . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → ω ≼ (ℵ‘𝐴))
21		domtr 8954	. . . . . . . . . . . . . . 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 9041	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → ((card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω → ¬ (ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴))))
26	25	con2d 134	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) ≺ (𝐵 ↑m (ℵ‘𝐴)) → ¬ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ ω))
27		cardidm 9883	. . . . . . . . . . . 12 ⊢ (card‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (card‘(𝐵 ↑m (ℵ‘𝐴)))
28		iscard3 10015	. . . . . . . . . . . . 13 ⊢ ((card‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (card‘(𝐵 ↑m (ℵ‘𝐴))) ↔ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
29		elun 4093	. . . . . . . . . . . . 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 849	. . . . . . . . . . . . 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 297	. . . . . . . . . . . 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 230	. . . . . . . . . . 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 9987	. . . . . . . . . . 11 ⊢ ℵ Fn On
35		fvelrnb 6900	. . . . . . . . . . 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 251	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → ∃𝑥 ∈ On (ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴)))))
38		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦)))
39	38	pwcfsdom 10506	. . . . . . . . . . 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 6840	. . . . . . . . . . . . 13 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
42	40, 41	oveq12d 7385	. . . . . . . . . . . 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 5098	. . . . . . . . . . 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 233	. . . . . . . . . 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 3134	. . . . . . . . 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 406	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (card‘(𝐵 ↑m (ℵ‘𝐴))) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
48		ensdomtr 9051	. . . . . . 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 588	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (𝐵 ↑m (ℵ‘𝐴)) ≺ ((card‘(𝐵 ↑m (ℵ‘𝐴))) ↑m (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
50		fvex 6853	. . . . . . . . 9 ⊢ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ∈ V
51	50	enref 8932	. . . . . . . 8 ⊢ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))
52		mapen 9079	. . . . . . . 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 693	. . . . . . 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 9081	. . . . . . . 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 1464	. . . . . . 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 8955	. . . . . 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 9049	. . . . . 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 587	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (𝐵 ↑m (ℵ‘𝐴)) ≺ (𝐵 ↑m ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))))
60	4	xpdom2 9010	. . . . . . . . . 10 ⊢ ((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) → ((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
61	16	biimpi 216	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ω ⊆ (ℵ‘𝐴))
62		infxpen 9936	. . . . . . . . . . 11 ⊢ (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
63	17, 61, 62	sylancr 588	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
64		domentr 8960	. . . . . . . . . 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 6408	. . . . . . . . . . 11 ⊢ suc 1o ≠ ∅
67		dom0 9043	. . . . . . . . . . 11 ⊢ (suc 1o ≼ ∅ ↔ suc 1o = ∅)
68	66, 67	nemtbir 3028	. . . . . . . . . 10 ⊢ ¬ suc 1o ≼ ∅
69		df-2o 8406	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
70	69	breq1i 5092	. . . . . . . . . . . . 13 ⊢ (2o ≼ 𝐵 ↔ suc 1o ≼ 𝐵)
71		breq2 5089	. . . . . . . . . . . . 13 ⊢ (𝐵 = ∅ → (suc 1o ≼ 𝐵 ↔ suc 1o ≼ ∅))
72	70, 71	bitrid 283	. . . . . . . . . . . 12 ⊢ (𝐵 = ∅ → (2o ≼ 𝐵 ↔ suc 1o ≼ ∅))
73	72	biimpcd 249	. . . . . . . . . . 11 ⊢ (2o ≼ 𝐵 → (𝐵 = ∅ → suc 1o ≼ ∅))
74	73	adantld 490	. . . . . . . . . 10 ⊢ (2o ≼ 𝐵 → ((((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅) → suc 1o ≼ ∅))
75	68, 74	mtoi 199	. . . . . . . . 9 ⊢ (2o ≼ 𝐵 → ¬ (((ℵ‘𝐴) × (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) = ∅ ∧ 𝐵 = ∅))
76		mapdom2 9086	. . . . . . . . 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 9041	. . . . . . . 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 457	. . . . . 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 136	. . . 4 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → ¬ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
83		domtri 10478	. . . . . 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 693	. . . . 5 ⊢ ((cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴) ↔ ¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
85	84	biimpri 228	. . . 4 ⊢ (¬ (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≼ (ℵ‘𝐴))
86	82, 85	nsyl2 141	. . 3 ⊢ ((𝐴 ∈ On ∧ 2o ≼ 𝐵) → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
87	86	ex 412	. 2 ⊢ (𝐴 ∈ On → (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))))
88		fndm 6601	. . . . . 6 ⊢ (ℵ Fn On → dom ℵ = On)
89	34, 88	ax-mp 5	. . . . 5 ⊢ dom ℵ = On
90	89	eleq2i 2828	. . . 4 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
91		ndmfv 6872	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
92	90, 91	sylnbir 331	. . 3 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
93		1n0 8423	. . . . . 6 ⊢ 1o ≠ ∅
94		1oex 8415	. . . . . . 7 ⊢ 1o ∈ V
95	94	0sdom 9046	. . . . . 6 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
96	93, 95	mpbir 231	. . . . 5 ⊢ ∅ ≺ 1o
97		id 22	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
98		oveq2 7375	. . . . . . . . . . 11 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑m (ℵ‘𝐴)) = (𝐵 ↑m ∅))
99		map0e 8830	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → (𝐵 ↑m ∅) = 1o)
100	54, 99	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐵 ↑m ∅) = 1o
101	98, 100	eqtrdi 2787	. . . . . . . . . 10 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑m (ℵ‘𝐴)) = 1o)
102	101	fveq2d 6844	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑m (ℵ‘𝐴))) = (card‘1o))
103		1onn 8576	. . . . . . . . . 10 ⊢ 1o ∈ ω
104		cardnn 9887	. . . . . . . . . 10 ⊢ (1o ∈ ω → (card‘1o) = 1o)
105	103, 104	ax-mp 5	. . . . . . . . 9 ⊢ (card‘1o) = 1o
106	102, 105	eqtrdi 2787	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑m (ℵ‘𝐴))) = 1o)
107	106	fveq2d 6844	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (cf‘1o))
108		df-1o 8405	. . . . . . . . 9 ⊢ 1o = suc ∅
109	108	fveq2i 6843	. . . . . . . 8 ⊢ (cf‘1o) = (cf‘suc ∅)
110		0elon 6378	. . . . . . . . 9 ⊢ ∅ ∈ On
111		cfsuc 10179	. . . . . . . . 9 ⊢ (∅ ∈ On → (cf‘suc ∅) = 1o)
112	110, 111	ax-mp 5	. . . . . . . 8 ⊢ (cf‘suc ∅) = 1o
113	109, 112	eqtri 2759	. . . . . . 7 ⊢ (cf‘1o) = 1o
114	107, 113	eqtrdi 2787	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = 1o)
115	97, 114	breq12d 5098	. . . . 5 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ↔ ∅ ≺ 1o))
116	96, 115	mpbiri 258	. . . 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 182	1 ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541   class class class wbr 5085   ↦ cmpt 5166   × cxp 5629  dom cdm 5631  ran crn 5632  Oncon0 6323  suc csuc 6325   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367  ωcom 7817  1_oc1o 8398  2_oc2o 8399   ↑_m cmap 8773   ≈ cen 8890   ≼ cdom 8891   ≺ csdm 8892  Fincfn 8893  harchar 9471  cardccrd 9859  ℵcale 9860  cfccf 9861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-inf2 9562  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-smo 8286  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-oi 9425  df-har 9472  df-card 9863  df-aleph 9864  df-cf 9865  df-acn 9866  df-ac 10038

This theorem is referenced by:  alephom  10508