Theorem cfsuc 10259

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 7813	. . 3 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2		cfval 10248	. . 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 220	. 2 ⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
4		cardsn 9974	. . . . . 6 ⊢ (𝐴 ∈ On → (card‘{𝐴}) = 1o)
5	4	eqcomd 2766	. . . . 5 ⊢ (𝐴 ∈ On → 1o = (card‘{𝐴}))
6		snidg 4621	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ {𝐴})
7		elsuci 6427	. . . . . . . . 9 ⊢ (𝑧 ∈ suc 𝐴 → (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴))
8		onelss 6400	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 ∈ 𝐴 → 𝑧 ⊆ 𝐴))
9		eqimss 3989	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → 𝑧 ⊆ 𝐴)
10	9	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 = 𝐴 → 𝑧 ⊆ 𝐴))
11	8, 10	jaod 873	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴) → 𝑧 ⊆ 𝐴))
12	7, 11	syl5 35	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → 𝑧 ⊆ 𝐴))
13		sseq2 3957	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝐴))
14	13	rspcev 3576	. . . . . . . 8 ⊢ ((𝐴 ∈ {𝐴} ∧ 𝑧 ⊆ 𝐴) → ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤)
15	6, 12, 14	syl6an 697	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑧 ∈ suc 𝐴 → ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
16	15	ralrimiv 3153	. . . . . 6 ⊢ (𝐴 ∈ On → ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤)
17		ssun2 4125	. . . . . . 7 ⊢ {𝐴} ⊆ (𝐴 ∪ {𝐴})
18		df-suc 6363	. . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
19	17, 18	sseqtrri 3980	. . . . . 6 ⊢ {𝐴} ⊆ suc 𝐴
20	16, 19	jctil 529	. . . . 5 ⊢ (𝐴 ∈ On → ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
21		snex 5404	. . . . . 6 ⊢ {𝐴} ∈ V
22		fveq2 6878	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (card‘𝑦) = (card‘{𝐴}))
23	22	eqeq2d 2771	. . . . . . 7 ⊢ (𝑦 = {𝐴} → (1o = (card‘𝑦) ↔ 1o = (card‘{𝐴})))
24		sseq1 3956	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (𝑦 ⊆ suc 𝐴 ↔ {𝐴} ⊆ suc 𝐴))
25		rexeq 3315	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
26	25	ralbidv 3185	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
27	24, 26	anbi12d 644	. . . . . . 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 644	. . . . . 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 3560	. . . . 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 596	. . . 4 ⊢ (𝐴 ∈ On → ∃𝑦(1o = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
31		1oex 8465	. . . . 5 ⊢ 1o ∈ V
32		eqeq1 2764	. . . . . . 7 ⊢ (𝑥 = 1o → (𝑥 = (card‘𝑦) ↔ 1o = (card‘𝑦)))
33	32	anbi1d 643	. . . . . 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 1954	. . . . 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 3633	. . . 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 237	. . 3 ⊢ (𝐴 ∈ On → 1o ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
37		el1o 8482	. . . . 5 ⊢ (𝑣 ∈ 1o ↔ 𝑣 = ∅)
38		eqcom 2767	. . . . . . . . . . . . . . 15 ⊢ (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
39		vex 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
40		onssnum 10043	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
41	39, 40	mpan 703	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
42		cardnueq0 9969	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ dom card → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
43	41, 42	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ On → ((card‘𝑦) = ∅ ↔ 𝑦 = ∅))
44	38, 43	bitrid 286	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ On → (∅ = (card‘𝑦) ↔ 𝑦 = ∅))
45	44	biimpa 482	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → 𝑦 = ∅)
46		rex0 4308	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
47	46	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ suc 𝐴 → ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
48	47	nrex 3090	. . . . . . . . . . . . . . 15 ⊢ ¬ ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
49		nsuceq0 6443	. . . . . . . . . . . . . . . 16 ⊢ suc 𝐴 ≠ ∅
50		r19.2z 4455	. . . . . . . . . . . . . . . 16 ⊢ ((suc 𝐴 ≠ ∅ ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤) → ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
51	49, 50	mpan 703	. . . . . . . . . . . . . . 15 ⊢ (∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 → ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
52	48, 51	mto 200	. . . . . . . . . . . . . 14 ⊢ ¬ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
53		rexeq 3315	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∅ → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
54	53	ralbidv 3185	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ∅ → (∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
55	52, 54	mtbiri 330	. . . . . . . . . . . . 13 ⊢ (𝑦 = ∅ → ¬ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)
56	45, 55	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)
57	56	intnand 494	. . . . . . . . . . 11 ⊢ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) → ¬ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
58		imnan 405	. . . . . . . . . . 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 233	. . . . . . . . . 10 ⊢ ¬ ((𝑦 ⊆ On ∧ ∅ = (card‘𝑦)) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
60		onsuc 7809	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
61		onss 7784	. . . . . . . . . . . . . . . . 17 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ⊆ On)
62		sstr 3939	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ⊆ On) → 𝑦 ⊆ On)
63	61, 62	sylan2 605	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ∈ On) → 𝑦 ⊆ On)
64	60, 63	sylan2 605	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ suc 𝐴 ∧ 𝐴 ∈ On) → 𝑦 ⊆ On)
65	64	ancoms 464	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑦 ⊆ suc 𝐴) → 𝑦 ⊆ On)
66	65	adantrr 730	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑦 ⊆ On)
67	66	3adant2 1149	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑦 ⊆ On)
68		simp2 1155	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ ∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → ∅ = (card‘𝑦))
69		simp3 1156	. . . . . . . . . . . 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 524	. . . . . . . . . . 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 1140	. . . . . . . . . 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 202	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ¬ (∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
73	72	nexdv 1969	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
74		0ex 5264	. . . . . . . . 9 ⊢ ∅ ∈ V
75		eqeq1 2764	. . . . . . . . . . 11 ⊢ (𝑥 = ∅ → (𝑥 = (card‘𝑦) ↔ ∅ = (card‘𝑦)))
76	75	anbi1d 643	. . . . . . . . . 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 1954	. . . . . . . . 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 3633	. . . . . . . 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 332	. . . . . . 7 ⊢ (𝐴 ∈ On → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
80	79	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ ∅ ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
81		eleq1 2848	. . . . . . 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 487	. . . . . 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 328	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝑣 = ∅) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
84	37, 83	sylan2b 606	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑣 ∈ 1o) → ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
85	84	ralrimiva 3154	. . 3 ⊢ (𝐴 ∈ On → ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
86		cardon 9949	. . . . . . . 8 ⊢ (card‘𝑦) ∈ On
87		eleq1 2848	. . . . . . . 8 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
88	86, 87	mpbiri 261	. . . . . . 7 ⊢ (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
89	88	adantr 486	. . . . . 6 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑥 ∈ On)
90	89	exlimiv 1963	. . . . 5 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑥 ∈ On)
91	90	abssi 4016	. . . 4 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On
92		oneqmini 6411	. . . 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 596	. 2 ⊢ (𝐴 ∈ On → 1o = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
95	3, 94	eqtr4d 2798	1 ⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = 1o)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  {csn 4584  ∩ cint 4907  dom cdm 5655  Oncon0 6357  suc csuc 6359  ‘cfv 6533  1_oc1o 8448  cardccrd 9940  cfccf 9942

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 7736

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-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 7370  df-ov 7416  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-1o 8455  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-card 9944  df-cf 9946

This theorem is used by:  cflim2  10265  cfpwsdom  10593  rankcf  10786