Theorem cfpwsdom 10501

Description: A corollary of Konig's Theorem konigth 10486. 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 7394	. . . . . . . . 9 ⊢ (𝐵 ↑m (ℵ‘𝐴)) ∈ V
2	1	cardid 10463	. . . . . . . 8 ⊢ (card‘(𝐵 ↑m (ℵ‘𝐴))) ≈ (𝐵 ↑m (ℵ‘𝐴))
3	2	ensymi 8945	. . . . . . 7 ⊢ (𝐵 ↑m (ℵ‘𝐴)) ≈ (card‘(𝐵 ↑m (ℵ‘𝐴)))
4		fvex 6848	. . . . . . . . . . . . . 14 ⊢ (ℵ‘𝐴) ∈ V
5	4	canth2 9062	. . . . . . . . . . . . 13 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
6	4	pw2en 9016	. . . . . . . . . . . . 13 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2o ↑m (ℵ‘𝐴))
7		sdomentr 9043	. . . . . . . . . . . . 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 9074	. . . . . . . . . . . 12 ⊢ (2o ≼ 𝐵 → (2o ↑m (ℵ‘𝐴)) ≼ (𝐵 ↑m (ℵ‘𝐴)))
10		sdomdomtr 9042	. . . . . . . . . . . 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 10481	. . . . . . . . . . . . . . . . 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 9568	. . . . . . . . . . . . . . . 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 9998	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		alephon 9985	. . . . . . . . . . . . . . . . 17 ⊢ (ℵ‘𝐴) ∈ On
18		ssdomg 8941	. . . . . . . . . . . . . . . . 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 8948	. . . . . . . . . . . . . . 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 9035	. . . . . . . . . . . . . 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 9877	. . . . . . . . . . . 12 ⊢ (card‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (card‘(𝐵 ↑m (ℵ‘𝐴)))
28		iscard3 10009	. . . . . . . . . . . . 13 ⊢ ((card‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (card‘(𝐵 ↑m (ℵ‘𝐴))) ↔ (card‘(𝐵 ↑m (ℵ‘𝐴))) ∈ (ω ∪ ran ℵ))
29		elun 4094	. . . . . . . . . . . . 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 9981	. . . . . . . . . . 11 ⊢ ℵ Fn On
35		fvelrnb 6895	. . . . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦))) = (𝑦 ∈ (cf‘(ℵ‘𝑥)) ↦ (har‘(𝑧‘𝑦)))
39	38	pwcfsdom 10500	. . . . . . . . . . 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 6835	. . . . . . . . . . . . 13 ⊢ ((ℵ‘𝑥) = (card‘(𝐵 ↑m (ℵ‘𝐴))) → (cf‘(ℵ‘𝑥)) = (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))))
42	40, 41	oveq12d 7379	. . . . . . . . . . . 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 5099	. . . . . . . . . . 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 3135	. . . . . . . . 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 9045	. . . . . . 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 6848	. . . . . . . . 9 ⊢ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ∈ V
51	50	enref 8926	. . . . . . . 8 ⊢ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) ≈ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))
52		mapen 9073	. . . . . . . 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 9075	. . . . . . . 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 8949	. . . . . 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 9043	. . . . . 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 9004	. . . . . . . . . 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 9930	. . . . . . . . . . 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 8954	. . . . . . . . . 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 6403	. . . . . . . . . . 11 ⊢ suc 1o ≠ ∅
67		dom0 9037	. . . . . . . . . . 11 ⊢ (suc 1o ≼ ∅ ↔ suc 1o = ∅)
68	66, 67	nemtbir 3029	. . . . . . . . . 10 ⊢ ¬ suc 1o ≼ ∅
69		df-2o 8400	. . . . . . . . . . . . . 14 ⊢ 2o = suc 1o
70	69	breq1i 5093	. . . . . . . . . . . . 13 ⊢ (2o ≼ 𝐵 ↔ suc 1o ≼ 𝐵)
71		breq2 5090	. . . . . . . . . . . . 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 9080	. . . . . . . . 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 9035	. . . . . . . 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 10472	. . . . . 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 6596	. . . . . 6 ⊢ (ℵ Fn On → dom ℵ = On)
89	34, 88	ax-mp 5	. . . . 5 ⊢ dom ℵ = On
90	89	eleq2i 2829	. . . 4 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
91		ndmfv 6867	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
92	90, 91	sylnbir 331	. . 3 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) = ∅)
93		1n0 8417	. . . . . 6 ⊢ 1o ≠ ∅
94		1oex 8409	. . . . . . 7 ⊢ 1o ∈ V
95	94	0sdom 9040	. . . . . 6 ⊢ (∅ ≺ 1o ↔ 1o ≠ ∅)
96	93, 95	mpbir 231	. . . . 5 ⊢ ∅ ≺ 1o
97		id 22	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
98		oveq2 7369	. . . . . . . . . . 11 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑m (ℵ‘𝐴)) = (𝐵 ↑m ∅))
99		map0e 8824	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V → (𝐵 ↑m ∅) = 1o)
100	54, 99	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐵 ↑m ∅) = 1o
101	98, 100	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((ℵ‘𝐴) = ∅ → (𝐵 ↑m (ℵ‘𝐴)) = 1o)
102	101	fveq2d 6839	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑m (ℵ‘𝐴))) = (card‘1o))
103		1onn 8570	. . . . . . . . . 10 ⊢ 1o ∈ ω
104		cardnn 9881	. . . . . . . . . 10 ⊢ (1o ∈ ω → (card‘1o) = 1o)
105	103, 104	ax-mp 5	. . . . . . . . 9 ⊢ (card‘1o) = 1o
106	102, 105	eqtrdi 2788	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (card‘(𝐵 ↑m (ℵ‘𝐴))) = 1o)
107	106	fveq2d 6839	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = (cf‘1o))
108		df-1o 8399	. . . . . . . . 9 ⊢ 1o = suc ∅
109	108	fveq2i 6838	. . . . . . . 8 ⊢ (cf‘1o) = (cf‘suc ∅)
110		0elon 6373	. . . . . . . . 9 ⊢ ∅ ∈ On
111		cfsuc 10173	. . . . . . . . 9 ⊢ (∅ ∈ On → (cf‘suc ∅) = 1o)
112	110, 111	ax-mp 5	. . . . . . . 8 ⊢ (cf‘suc ∅) = 1o
113	109, 112	eqtri 2760	. . . . . . 7 ⊢ (cf‘1o) = 1o
114	107, 113	eqtrdi 2788	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴)))) = 1o)
115	97, 114	breq12d 5099	. . . . 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 2933  ∃wrex 3062  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542   class class class wbr 5086   ↦ cmpt 5167   × cxp 5623  dom cdm 5625  ran crn 5626  Oncon0 6318  suc csuc 6320   Fn wfn 6488  ‘cfv 6493  (class class class)co 7361  ωcom 7811  1_oc1o 8392  2_oc2o 8393   ↑_m cmap 8767   ≈ cen 8884   ≼ cdom 8885   ≺ csdm 8886  Fincfn 8887  harchar 9465  cardccrd 9853  ℵcale 9854  cfccf 9855

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556  ax-ac2 10379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-smo 8280  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-oi 9419  df-har 9466  df-card 9857  df-aleph 9858  df-cf 9859  df-acn 9860  df-ac 10032

This theorem is referenced by:  alephom  10502