Theorem cfsuc 10106

Description: Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.)

Assertion

Ref	Expression
cfsuc	⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = 1o)

Proof of Theorem cfsuc

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

Step	Hyp	Ref	Expression
1		sucelon 7722	. . 3 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2		cfval 10096	. . 3 ⊢ (suc 𝐴 ∈ On → (cf‘suc 𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
3	1, 2	sylbi 216	. 2 ⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
4		cardsn 9818	. . . . . 6 ⊢ (𝐴 ∈ On → (card‘{𝐴}) = 1o)
5	4	eqcomd 2742	. . . . 5 ⊢ (𝐴 ∈ On → 1o = (card‘{𝐴}))
6		snidg 4606	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ {𝐴})
7		elsuci 6362	. . . . . . . . 9 ⊢ (𝑧 ∈ suc 𝐴 → (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴))
8		onelss 6338	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 ∈ 𝐴 → 𝑧 ⊆ 𝐴))
9		eqimss 3987	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → 𝑧 ⊆ 𝐴)
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 = 𝐴 → 𝑧 ⊆ 𝐴))
11	8, 10	jaod 856	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴) → 𝑧 ⊆ 𝐴))
12	7, 11	syl5 34	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → 𝑧 ⊆ 𝐴))
13		sseq2 3957	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝐴))
14	13	rspcev 3570	. . . . . . . 8 ⊢ ((𝐴 ∈ {𝐴} ∧ 𝑧 ⊆ 𝐴) → ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤)
15	6, 12, 14	syl6an 681	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
16	15	ralrimiv 3138	. . . . . 6 ⊢ (𝐴 ∈ On → ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤)
17		ssun2 4119	. . . . . . 7 ⊢ {𝐴} ⊆ (𝐴 ∪ {𝐴})
18		df-suc 6302	. . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
19	17, 18	sseqtrri 3968	. . . . . 6 ⊢ {𝐴} ⊆ suc 𝐴
20	16, 19	jctil 520	. . . . 5 ⊢ (𝐴 ∈ On → ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
21		snex 5373	. . . . . 6 ⊢ {𝐴} ∈ V
22		fveq2 6819	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (card‘𝑦) = (card‘{𝐴}))
23	22	eqeq2d 2747	. . . . . . 7 ⊢ (𝑦 = {𝐴} → (1o = (card‘𝑦) ↔ 1o = (card‘{𝐴})))
24		sseq1 3956	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (𝑦 ⊆ suc 𝐴 ↔ {𝐴} ⊆ suc 𝐴))
25		rexeq 3306	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
26	25	ralbidv 3170	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
27	24, 26	anbi12d 631	. . . . . . 7 ⊢ (𝑦 = {𝐴} → ((𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤)))
28	23, 27	anbi12d 631	. . . . . 6 ⊢ (𝑦 = {𝐴} → ((1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (1o = (card‘{𝐴}) ∧ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))))
29	21, 28	spcev 3554	. . . . 5 ⊢ ((1o = (card‘{𝐴}) ∧ ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤)) → ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
30	5, 20, 29	syl2anc 584	. . . 4 ⊢ (𝐴 ∈ On → ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
31		1oex 8369	. . . . 5 ⊢ 1o ∈ V
32		eqeq1 2740	. . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = (card‘𝑦) ↔ 1o = (card‘𝑦)))
33	32	anbi1d 630	. . . . . 6 ⊢ (𝑥 = 1o → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
34	33	exbidv 1923	. . . . 5 ⊢ (𝑥 = 1o → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
35	31, 34	elab 3619	. . . 4 ⊢ (1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ↔ ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
36	30, 35	sylibr 233	. . 3 ⊢ (𝐴 ∈ On → 1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
37		el1o 8388	. . . . 5 ⊢ (𝑣 ∈ 1o ↔ 𝑣 = ∅)
38		eqcom 2743	. . . . . . . . . . . . . . 15 ⊢ (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
39		vex 3445	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
40		onssnum 9889	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
41	39, 40	mpan 687	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
42		cardnueq0 9813	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
43	41, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
44	38, 43	bitrid 282	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
45	44	biimpa 477	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
46		rex0 4303	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
47	46	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ suc 𝐴 → ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
48	47	nrex 3074	. . . . . . . . . . . . . . 15 ⊢ ¬ ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
49		nsuceq0 6378	. . . . . . . . . . . . . . . 16 ⊢ suc 𝐴 ≠ ∅
50		r19.2z 4438	. . . . . . . . . . . . . . . 16 ⊢ ((suc 𝐴 ≠ ∅ ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤) → ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
51	49, 50	mpan 687	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 → ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
52	48, 51	mto 196	. . . . . . . . . . . . . 14 ⊢ ¬ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
53		rexeq 3306	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∅ → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
54	53	ralbidv 3170	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
55	52, 54	mtbiri 326	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → ¬ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)
56	45, 55	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)
57	56	intnand 489	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
58		imnan 400	. . . . . . . . . . 11 ⊢ (((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
59	57, 58	mpbi 229	. . . . . . . . . 10 ⊢ ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
60		suceloni 7716	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
61		onss 7689	. . . . . . . . . . . . . . . . 17 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ⊆ On)
62		sstr 3939	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ⊆ On) → 𝑦 ⊆ On)
63	61, 62	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ∈ On) → 𝑦 ⊆ On)
64	60, 63	sylan2 593	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ suc 𝐴 ∧ 𝐴 ∈ On) → 𝑦 ⊆ On)
65	64	ancoms 459	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ suc 𝐴) → 𝑦 ⊆ On)
66	65	adantrr 714	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑦 ⊆ On)
67	66	3adant2 1130	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑦 ⊆ On)
68		simp2 1136	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → ∅ = (card‘𝑦))
69		simp3 1137	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
70	67, 68, 69	jca31 515	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
71	70	3expib 1121	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
72	59, 71	mtoi 198	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ¬ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
73	72	nexdv 1938	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
74		0ex 5248	. . . . . . . . 9 ⊢ ∅ ∈ V
75		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
76	75	anbi1d 630	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
77	76	exbidv 1923	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
78	74, 77	elab 3619	. . . . . . . 8 ⊢ (∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ↔ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
79	73, 78	sylnibr 328	. . . . . . 7 ⊢ (𝐴 ∈ On → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
80	79	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
81		eleq1 2824	. . . . . . 7 ⊢ (𝑣 = ∅ → (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}))
82	81	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑣 = ∅) → (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ↔ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}))
83	80, 82	mtbird 324	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
84	37, 83	sylan2b 594	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑣 ∈ 1o) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
85	84	ralrimiva 3139	. . 3 ⊢ (𝐴 ∈ On → ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
86		cardon 9793	. . . . . . . 8 ⊢ (card‘𝑦) ∈ On
87		eleq1 2824	. . . . . . . 8 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
88	86, 87	mpbiri 257	. . . . . . 7 ⊢ (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
89	88	adantr 481	. . . . . 6 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑥 ∈ On)
90	89	exlimiv 1932	. . . . 5 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑥 ∈ On)
91	90	abssi 4014	. . . 4 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On
92		oneqmini 6347	. . . 4 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On → ((1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ∧ ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}) → 1o = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}))
93	91, 92	ax-mp 5	. . 3 ⊢ ((1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ∧ ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}) → 1o = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
94	36, 85, 93	syl2anc 584	. 2 ⊢ (𝐴 ∈ On → 1o = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
95	3, 94	eqtr4d 2779	1 ⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = 1o)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1540  ∃wex 1780   ∈ wcel 2105  {cab 2713   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3441   ∪ cun 3895   ⊆ wss 3897  ∅c0 4268  {csn 4572  ∩ cint 4893  dom cdm 5614  Oncon0 6296  suc csuc 6298  ‘cfv 6473  1_oc1o 8352  cardccrd 9784  cfccf 9786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-se 5570  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-isom 6482  df-riota 7286  df-ov 7332  df-om 7773  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-1o 8359  df-er 8561  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-card 9788  df-cf 9790

This theorem is referenced by:  cflim2  10112  cfpwsdom  10433  rankcf  10626