Theorem cfsuc 10151

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 7750	. . 3 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
2		cfval 10141	. . 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 9865	. . . . . 6 ⊢ (𝐴 ∈ On → (card‘{𝐴}) = 1o)
5	4	eqcomd 2735	. . . . 5 ⊢ (𝐴 ∈ On → 1o = (card‘{𝐴}))
6		snidg 4612	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ∈ {𝐴})
7		elsuci 6376	. . . . . . . . 9 ⊢ (𝑧 ∈ suc 𝐴 → (𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴))
8		onelss 6349	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑧 ∈ 𝐴 → 𝑧 ⊆ 𝐴))
9		eqimss 3994	. . . . . . . . . . 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 3962	. . . . . . . . 9 ⊢ (𝑤 = 𝐴 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝐴))
14	13	rspcev 3577	. . . . . . . 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 3120	. . . . . 6 ⊢ (𝐴 ∈ On → ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤)
17		ssun2 4130	. . . . . . 7 ⊢ {𝐴} ⊆ (𝐴 ∪ {𝐴})
18		df-suc 6313	. . . . . . 7 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
19	17, 18	sseqtrri 3985	. . . . . 6 ⊢ {𝐴} ⊆ suc 𝐴
20	16, 19	jctil 519	. . . . 5 ⊢ (𝐴 ∈ On → ({𝐴} ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
21		snex 5375	. . . . . 6 ⊢ {𝐴} ∈ V
22		fveq2 6822	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (card‘𝑦) = (card‘{𝐴}))
23	22	eqeq2d 2740	. . . . . . 7 ⊢ (𝑦 = {𝐴} → (1o = (card‘𝑦) ↔ 1o = (card‘{𝐴})))
24		sseq1 3961	. . . . . . . 8 ⊢ (𝑦 = {𝐴} → (𝑦 ⊆ suc 𝐴 ↔ {𝐴} ⊆ suc 𝐴))
25		rexeq 3285	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ {𝐴}𝑧 ⊆ 𝑤))
26	25	ralbidv 3152	. . . . . . . 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 3561	. . . . 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 8398	. . . . 5 ⊢ 1o ∈ V
32		eqeq1 2733	. . . . . . 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 1921	. . . . 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 3635	. . . 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 8413	. . . . 5 ⊢ (𝑣 ∈ 1o ↔ 𝑣 = ∅)
38		eqcom 2736	. . . . . . . . . . . . . . 15 ⊢ (∅ = (card‘𝑦) ↔ (card‘𝑦) = ∅)
39		vex 3440	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
40		onssnum 9934	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
41	39, 40	mpan 690	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
42		cardnueq0 9860	. . . . . . . . . . . . . . . 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 4311	. . . . . . . . . . . . . . . . 17 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
47	46	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ suc 𝐴 → ¬ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤)
48	47	nrex 3057	. . . . . . . . . . . . . . 15 ⊢ ¬ ∃𝑧 ∈ suc 𝐴∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤
49		nsuceq0 6392	. . . . . . . . . . . . . . . 16 ⊢ suc 𝐴 ≠ ∅
50		r19.2z 4446	. . . . . . . . . . . . . . . 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 3285	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ∅ → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ ∅ 𝑧 ⊆ 𝑤))
54	53	ralbidv 3152	. . . . . . . . . . . . . 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 7746	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
61		onss 7721	. . . . . . . . . . . . . . . . 17 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ⊆ On)
62		sstr 3944	. . . . . . . . . . . . . . . . 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 1936	. . . . . . . 8 ⊢ (𝐴 ∈ On → ¬ ∃𝑦(∅ = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
74		0ex 5246	. . . . . . . . 9 ⊢ ∅ ∈ V
75		eqeq1 2733	. . . . . . . . . . 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 1921	. . . . . . . . 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 3635	. . . . . . . 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 2816	. . . . . . 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 3121	. . 3 ⊢ (𝐴 ∈ On → ∀𝑣 ∈ 1o ¬ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
86		cardon 9840	. . . . . . . 8 ⊢ (card‘𝑦) ∈ On
87		eleq1 2816	. . . . . . . 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 1930	. . . . 5 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → 𝑥 ∈ On)
91	90	abssi 4021	. . . 4 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ suc 𝐴 ∧ ∀𝑧 ∈ suc 𝐴∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On
92		oneqmini 6360	. . . 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 2767	1 ⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = 1o)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  {csn 4577  ∩ cint 4896  dom cdm 5619  Oncon0 6307  suc csuc 6309  ‘cfv 6482  1_oc1o 8381  cardccrd 9831  cfccf 9833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-card 9835  df-cf 9837

This theorem is referenced by:  cflim2  10157  cfpwsdom  10478  rankcf  10671