Theorem cfsuc 10143

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		onsucb 7742	. . 3 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2		cfval 10133	. . 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 217	. 2 ⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
4		cardsn 9857	. . . . . 6 ⊢ (𝐴 ∈ On → (card‘{𝐴}) = 1o)
5	4	eqcomd 2737	. . . . 5 ⊢ (𝐴 ∈ On → 1o = (card‘{𝐴}))
6		snidg 4608	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ {𝐴})
7		elsuci 6370	. . . . . . . . 9 ⊢ (𝑧 ∈ suc 𝐴 → (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴))
8		onelss 6343	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 ∈ 𝐴 → 𝑧 ⊆ 𝐴))
9		eqimss 3988	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → 𝑧 ⊆ 𝐴)
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 = 𝐴 → 𝑧 ⊆ 𝐴))
11	8, 10	jaod 859	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴) → 𝑧 ⊆ 𝐴))
12	7, 11	syl5 34	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → 𝑧 ⊆ 𝐴))
13		sseq2 3956	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝐴))
14	13	rspcev 3572	. . . . . . . 8 ⊢ ((𝐴 ∈ {𝐴} ∧ 𝑧 ⊆ 𝐴) → ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤)
15	6, 12, 14	syl6an 684	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
16	15	ralrimiv 3123	. . . . . 6 ⊢ (𝐴 ∈ On → ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤)
17		ssun2 4124	. . . . . . 7 ⊢ {𝐴} ⊆ (𝐴 ∪ {𝐴})
18		df-suc 6307	. . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
19	17, 18	sseqtrri 3979	. . . . . 6 ⊢ {𝐴} ⊆ suc 𝐴
20	16, 19	jctil 519	. . . . 5 ⊢ (𝐴 ∈ On → ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
21		snex 5369	. . . . . 6 ⊢ {𝐴} ∈ V
22		fveq2 6817	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (card‘𝑦) = (card‘{𝐴}))
23	22	eqeq2d 2742	. . . . . . 7 ⊢ (𝑦 = {𝐴} → (1o = (card‘𝑦) ↔ 1o = (card‘{𝐴})))
24		sseq1 3955	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (𝑦 ⊆ suc 𝐴 ↔ {𝐴} ⊆ suc 𝐴))
25		rexeq 3288	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
26	25	ralbidv 3155	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
27	24, 26	anbi12d 632	. . . . . . 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 632	. . . . . 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 3556	. . . . 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 8390	. . . . 5 ⊢ 1o ∈ V
32		eqeq1 2735	. . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = (card‘𝑦) ↔ 1o = (card‘𝑦)))
33	32	anbi1d 631	. . . . . 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 1922	. . . . 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 3630	. . . 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 234	. . 3 ⊢ (𝐴 ∈ On → 1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
37		el1o 8405	. . . . 5 ⊢ (𝑣 ∈ 1o ↔ 𝑣 = ∅)
38		eqcom 2738	. . . . . . . . . . . . . . 15 ⊢ (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
39		vex 3440	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
40		onssnum 9926	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
41	39, 40	mpan 690	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
42		cardnueq0 9852	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
43	41, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
44	38, 43	bitrid 283	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
45	44	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
46		rex0 4305	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
47	46	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ suc 𝐴 → ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
48	47	nrex 3060	. . . . . . . . . . . . . . 15 ⊢ ¬ ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
49		nsuceq0 6386	. . . . . . . . . . . . . . . 16 ⊢ suc 𝐴 ≠ ∅
50		r19.2z 4440	. . . . . . . . . . . . . . . 16 ⊢ ((suc 𝐴 ≠ ∅ ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤) → ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
51	49, 50	mpan 690	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 → ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
52	48, 51	mto 197	. . . . . . . . . . . . . 14 ⊢ ¬ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
53		rexeq 3288	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∅ → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
54	53	ralbidv 3155	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
55	52, 54	mtbiri 327	. . . . . . . . . . . . 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 488	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
58		imnan 399	. . . . . . . . . . 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 230	. . . . . . . . . 10 ⊢ ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
60		onsuc 7738	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
61		onss 7713	. . . . . . . . . . . . . . . . 17 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ⊆ On)
62		sstr 3938	. . . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ suc 𝐴) → 𝑦 ⊆ On)
66	65	adantrr 717	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑦 ⊆ On)
67	66	3adant2 1131	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑦 ⊆ On)
68		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → ∅ = (card‘𝑦))
69		simp3 1138	. . . . . . . . . . . 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 514	. . . . . . . . . . 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 1122	. . . . . . . . . 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 199	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ¬ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
73	72	nexdv 1937	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
74		0ex 5240	. . . . . . . . 9 ⊢ ∅ ∈ V
75		eqeq1 2735	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
76	75	anbi1d 631	. . . . . . . . . 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 1922	. . . . . . . . 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 3630	. . . . . . . 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 329	. . . . . . 7 ⊢ (𝐴 ∈ On → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
80	79	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
81		eleq1 2819	. . . . . . 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 481	. . . . . 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 325	. . . . 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 3124	. . 3 ⊢ (𝐴 ∈ On → ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
86		cardon 9832	. . . . . . . 8 ⊢ (card‘𝑦) ∈ On
87		eleq1 2819	. . . . . . . 8 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
88	86, 87	mpbiri 258	. . . . . . 7 ⊢ (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
89	88	adantr 480	. . . . . 6 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑥 ∈ On)
90	89	exlimiv 1931	. . . . 5 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑥 ∈ On)
91	90	abssi 4015	. . . 4 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On
92		oneqmini 6354	. . . 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 2769	1 ⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = 1o)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∪ cun 3895   ⊆ wss 3897  ∅c0 4278  {csn 4571  ∩ cint 4892  dom cdm 5611  Oncon0 6301  suc csuc 6303  ‘cfv 6476  1_oc1o 8373  cardccrd 9823  cfccf 9825

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-1o 8380  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-card 9827  df-cf 9829

This theorem is referenced by:  cflim2  10149  cfpwsdom  10470  rankcf  10663