Theorem cflim2 9085

Description: The cofinality function is a limit ordinal iff its argument is. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)

Hypothesis

Ref	Expression
cflim2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cflim2	⊢ (Lim 𝐴 ↔ Lim (cf‘𝐴))

Proof of Theorem cflim2

Dummy variables 𝑠 𝑦 𝑥 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabid 3116	. . . . . . 7 ⊢ (𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ ∪ 𝑦 = 𝐴))
2		selpw 4165	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
3		limord 5784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Lim 𝐴 → Ord 𝐴)
4		ordsson 6989	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Ord 𝐴 → 𝐴 ⊆ On)
5		sstr 3611	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ On) → 𝑦 ⊆ On)
6	5	expcom 451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ On → (𝑦 ⊆ 𝐴 → 𝑦 ⊆ On))
7	3, 4, 6	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim 𝐴 → (𝑦 ⊆ 𝐴 → 𝑦 ⊆ On))
8	7	imp 445	. . . . . . . . . . . . . . . . . 18 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ On)
9	8	3adant3 1081	. . . . . . . . . . . . . . . . 17 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) → 𝑦 ⊆ On)
10		ssel2 3598	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ⊆ On ∧ 𝑠 ∈ 𝑦) → 𝑠 ∈ On)
11		eloni 5733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ On → Ord 𝑠)
12		ordirr 5741	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑠 → ¬ 𝑠 ∈ 𝑠)
13	10, 11, 12	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ⊆ On ∧ 𝑠 ∈ 𝑦) → ¬ 𝑠 ∈ 𝑠)
14		ssel 3597	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ⊆ 𝑠 → (𝑠 ∈ 𝑦 → 𝑠 ∈ 𝑠))
15	14	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ 𝑦 → (𝑦 ⊆ 𝑠 → 𝑠 ∈ 𝑠))
16	15	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ⊆ On ∧ 𝑠 ∈ 𝑦) → (𝑦 ⊆ 𝑠 → 𝑠 ∈ 𝑠))
17	13, 16	mtod 189	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ On ∧ 𝑠 ∈ 𝑦) → ¬ 𝑦 ⊆ 𝑠)
18	9, 17	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ 𝑠 ∈ 𝑦) → ¬ 𝑦 ⊆ 𝑠)
19		simpl2 1065	. . . . . . . . . . . . . . . . 17 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ 𝑠 ∈ 𝑦) → 𝑦 ⊆ 𝐴)
20		sstr 3611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑠) → 𝑦 ⊆ 𝑠)
21	19, 20	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ ((((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ 𝑠 ∈ 𝑦) ∧ 𝐴 ⊆ 𝑠) → 𝑦 ⊆ 𝑠)
22	18, 21	mtand 691	. . . . . . . . . . . . . . 15 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ 𝑠 ∈ 𝑦) → ¬ 𝐴 ⊆ 𝑠)
23		simpl3 1066	. . . . . . . . . . . . . . . 16 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ 𝑠 ∈ 𝑦) → ∪ 𝑦 = 𝐴)
24	23	sseq1d 3632	. . . . . . . . . . . . . . 15 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ 𝑠 ∈ 𝑦) → (∪ 𝑦 ⊆ 𝑠 ↔ 𝐴 ⊆ 𝑠))
25	22, 24	mtbird 315	. . . . . . . . . . . . . 14 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ 𝑠 ∈ 𝑦) → ¬ ∪ 𝑦 ⊆ 𝑠)
26		unissb 4469	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑦 ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝑦 𝑡 ⊆ 𝑠)
27	25, 26	sylnib 318	. . . . . . . . . . . . 13 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ 𝑠 ∈ 𝑦) → ¬ ∀𝑡 ∈ 𝑦 𝑡 ⊆ 𝑠)
28	27	nrexdv 3001	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) → ¬ ∃𝑠 ∈ 𝑦 ∀𝑡 ∈ 𝑦 𝑡 ⊆ 𝑠)
29		ssel 3597	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ⊆ On → (𝑠 ∈ 𝑦 → 𝑠 ∈ On))
30		ssel 3597	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ⊆ On → (𝑡 ∈ 𝑦 → 𝑡 ∈ On))
31		ontri1 5757	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑡 ∈ On ∧ 𝑠 ∈ On) → (𝑡 ⊆ 𝑠 ↔ ¬ 𝑠 ∈ 𝑡))
32	31	ancoms 469	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡 ⊆ 𝑠 ↔ ¬ 𝑠 ∈ 𝑡))
33		vex 3203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑡 ∈ V
34		vex 3203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑠 ∈ V
35	33, 34	brcnv 5305	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡◡ E 𝑠 ↔ 𝑠 E 𝑡)
36		epel 5032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 E 𝑡 ↔ 𝑠 ∈ 𝑡)
37	35, 36	bitri 264	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡◡ E 𝑠 ↔ 𝑠 ∈ 𝑡)
38	37	notbii 310	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑡◡ E 𝑠 ↔ ¬ 𝑠 ∈ 𝑡)
39	32, 38	syl6bbr 278	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡 ⊆ 𝑠 ↔ ¬ 𝑡◡ E 𝑠))
40	39	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ⊆ On → ((𝑠 ∈ On ∧ 𝑡 ∈ On) → (𝑡 ⊆ 𝑠 ↔ ¬ 𝑡◡ E 𝑠)))
41	29, 30, 40	syl2and 500	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ On → ((𝑠 ∈ 𝑦 ∧ 𝑡 ∈ 𝑦) → (𝑡 ⊆ 𝑠 ↔ ¬ 𝑡◡ E 𝑠)))
42	41	impl 650	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ⊆ On ∧ 𝑠 ∈ 𝑦) ∧ 𝑡 ∈ 𝑦) → (𝑡 ⊆ 𝑠 ↔ ¬ 𝑡◡ E 𝑠))
43	42	ralbidva 2985	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ On ∧ 𝑠 ∈ 𝑦) → (∀𝑡 ∈ 𝑦 𝑡 ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝑦 ¬ 𝑡◡ E 𝑠))
44	43	rexbidva 3049	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ On → (∃𝑠 ∈ 𝑦 ∀𝑡 ∈ 𝑦 𝑡 ⊆ 𝑠 ↔ ∃𝑠 ∈ 𝑦 ∀𝑡 ∈ 𝑦 ¬ 𝑡◡ E 𝑠))
45	9, 44	syl 17	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) → (∃𝑠 ∈ 𝑦 ∀𝑡 ∈ 𝑦 𝑡 ⊆ 𝑠 ↔ ∃𝑠 ∈ 𝑦 ∀𝑡 ∈ 𝑦 ¬ 𝑡◡ E 𝑠))
46	28, 45	mtbid 314	. . . . . . . . . . 11 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) → ¬ ∃𝑠 ∈ 𝑦 ∀𝑡 ∈ 𝑦 ¬ 𝑡◡ E 𝑠)
47		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
48	47	a1i 11	. . . . . . . . . . . 12 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ V)
49		epweon 6983	. . . . . . . . . . . . . . . . . . 19 ⊢ E We On
50		wess 5101	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ⊆ On → ( E We On → E We 𝑦))
51	49, 50	mpi 20	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ⊆ On → E We 𝑦)
52		weso 5105	. . . . . . . . . . . . . . . . . 18 ⊢ ( E We 𝑦 → E Or 𝑦)
53	51, 52	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ⊆ On → E Or 𝑦)
54		cnvso 5674	. . . . . . . . . . . . . . . . 17 ⊢ ( E Or 𝑦 ↔ ◡ E Or 𝑦)
55	53, 54	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ⊆ On → ◡ E Or 𝑦)
56	55	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → ◡ E Or 𝑦)
57		onssnum 8863	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ V ∧ 𝑦 ⊆ On) → 𝑦 ∈ dom card)
58	47, 57	mpan 706	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ⊆ On → 𝑦 ∈ dom card)
59		cardid2 8779	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ dom card → (card‘𝑦) ≈ 𝑦)
60		ensym 8005	. . . . . . . . . . . . . . . . . 18 ⊢ ((card‘𝑦) ≈ 𝑦 → 𝑦 ≈ (card‘𝑦))
61	58, 59, 60	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ⊆ On → 𝑦 ≈ (card‘𝑦))
62		nnsdom 8551	. . . . . . . . . . . . . . . . 17 ⊢ ((card‘𝑦) ∈ ω → (card‘𝑦) ≺ ω)
63		ensdomtr 8096	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ≈ (card‘𝑦) ∧ (card‘𝑦) ≺ ω) → 𝑦 ≺ ω)
64	61, 62, 63	syl2an 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ≺ ω)
65		isfinite 8549	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ Fin ↔ 𝑦 ≺ ω)
66	64, 65	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → 𝑦 ∈ Fin)
67		wofi 8209	. . . . . . . . . . . . . . 15 ⊢ ((◡ E Or 𝑦 ∧ 𝑦 ∈ Fin) → ◡ E We 𝑦)
68	56, 66, 67	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ⊆ On ∧ (card‘𝑦) ∈ ω) → ◡ E We 𝑦)
69	9, 68	sylan 488	. . . . . . . . . . . . 13 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → ◡ E We 𝑦)
70		wefr 5104	. . . . . . . . . . . . 13 ⊢ (◡ E We 𝑦 → ◡ E Fr 𝑦)
71	69, 70	syl 17	. . . . . . . . . . . 12 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → ◡ E Fr 𝑦)
72		ssid 3624	. . . . . . . . . . . . 13 ⊢ 𝑦 ⊆ 𝑦
73	72	a1i 11	. . . . . . . . . . . 12 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ⊆ 𝑦)
74		unieq 4444	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∪ ∅)
75		uni0 4465	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ ∅ = ∅
76	74, 75	syl6eq 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∅)
77		eqeq1 2626	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑦 = 𝐴 → (∪ 𝑦 = ∅ ↔ 𝐴 = ∅))
78	76, 77	syl5ib 234	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑦 = 𝐴 → (𝑦 = ∅ → 𝐴 = ∅))
79		nlim0 5783	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ Lim ∅
80		limeq 5735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅))
81	79, 80	mtbiri 317	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 = ∅ → ¬ Lim 𝐴)
82	78, 81	syl6 35	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑦 = 𝐴 → (𝑦 = ∅ → ¬ Lim 𝐴))
83	82	necon2ad 2809	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑦 = 𝐴 → (Lim 𝐴 → 𝑦 ≠ ∅))
84	83	impcom 446	. . . . . . . . . . . . . 14 ⊢ ((Lim 𝐴 ∧ ∪ 𝑦 = 𝐴) → 𝑦 ≠ ∅)
85	84	3adant2 1080	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) → 𝑦 ≠ ∅)
86	85	adantr 481	. . . . . . . . . . . 12 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → 𝑦 ≠ ∅)
87		fri 5076	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ V ∧ ◡ E Fr 𝑦) ∧ (𝑦 ⊆ 𝑦 ∧ 𝑦 ≠ ∅)) → ∃𝑠 ∈ 𝑦 ∀𝑡 ∈ 𝑦 ¬ 𝑡◡ E 𝑠)
88	48, 71, 73, 86, 87	syl22anc 1327	. . . . . . . . . . 11 ⊢ (((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) ∧ (card‘𝑦) ∈ ω) → ∃𝑠 ∈ 𝑦 ∀𝑡 ∈ 𝑦 ¬ 𝑡◡ E 𝑠)
89	46, 88	mtand 691	. . . . . . . . . 10 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) → ¬ (card‘𝑦) ∈ ω)
90		cardon 8770	. . . . . . . . . . 11 ⊢ (card‘𝑦) ∈ On
91		eloni 5733	. . . . . . . . . . 11 ⊢ ((card‘𝑦) ∈ On → Ord (card‘𝑦))
92		ordom 7074	. . . . . . . . . . . 12 ⊢ Ord ω
93		ordtri1 5756	. . . . . . . . . . . 12 ⊢ ((Ord ω ∧ Ord (card‘𝑦)) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
94	92, 93	mpan 706	. . . . . . . . . . 11 ⊢ (Ord (card‘𝑦) → (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω))
95	90, 91, 94	mp2b 10	. . . . . . . . . 10 ⊢ (ω ⊆ (card‘𝑦) ↔ ¬ (card‘𝑦) ∈ ω)
96	89, 95	sylibr 224	. . . . . . . . 9 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴 ∧ ∪ 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
97	2, 96	syl3an2b 1363	. . . . . . . 8 ⊢ ((Lim 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴 ∧ ∪ 𝑦 = 𝐴) → ω ⊆ (card‘𝑦))
98	97	3expb 1266	. . . . . . 7 ⊢ ((Lim 𝐴 ∧ (𝑦 ∈ 𝒫 𝐴 ∧ ∪ 𝑦 = 𝐴)) → ω ⊆ (card‘𝑦))
99	1, 98	sylan2b 492	. . . . . 6 ⊢ ((Lim 𝐴 ∧ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}) → ω ⊆ (card‘𝑦))
100	99	ralrimiva 2966	. . . . 5 ⊢ (Lim 𝐴 → ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
101		ssiin 4570	. . . . 5 ⊢ (ω ⊆ ∩ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴} (card‘𝑦) ↔ ∀𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}ω ⊆ (card‘𝑦))
102	100, 101	sylibr 224	. . . 4 ⊢ (Lim 𝐴 → ω ⊆ ∩ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴} (card‘𝑦))
103		cflim2.1	. . . . 5 ⊢ 𝐴 ∈ V
104	103	cflim3 9084	. . . 4 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴} (card‘𝑦))
105	102, 104	sseqtr4d 3642	. . 3 ⊢ (Lim 𝐴 → ω ⊆ (cf‘𝐴))
106		fvex 6201	. . . . . . 7 ⊢ (card‘𝑦) ∈ V
107	106	dfiin2 4555	. . . . . 6 ⊢ ∩ 𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴} (card‘𝑦) = ∩ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)}
108	104, 107	syl6eq 2672	. . . . 5 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
109		cardlim 8798	. . . . . . . . 9 ⊢ (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))
110		sseq2 3627	. . . . . . . . . 10 ⊢ (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ ω ⊆ (card‘𝑦)))
111		limeq 5735	. . . . . . . . . 10 ⊢ (𝑥 = (card‘𝑦) → (Lim 𝑥 ↔ Lim (card‘𝑦)))
112	110, 111	bibi12d 335	. . . . . . . . 9 ⊢ (𝑥 = (card‘𝑦) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (card‘𝑦) ↔ Lim (card‘𝑦))))
113	109, 112	mpbiri 248	. . . . . . . 8 ⊢ (𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
114	113	rexlimivw 3029	. . . . . . 7 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦) → (ω ⊆ 𝑥 ↔ Lim 𝑥))
115	114	ss2abi 3674	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)}
116		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ On ↔ (card‘𝑦) ∈ On))
117	90, 116	mpbiri 248	. . . . . . . . 9 ⊢ (𝑥 = (card‘𝑦) → 𝑥 ∈ On)
118	117	rexlimivw 3029	. . . . . . . 8 ⊢ (∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦) → 𝑥 ∈ On)
119	118	abssi 3677	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ On
120		fvex 6201	. . . . . . . . 9 ⊢ (cf‘𝐴) ∈ V
121	108, 120	syl6eqelr 2710	. . . . . . . 8 ⊢ (Lim 𝐴 → ∩ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
122		intex 4820	. . . . . . . 8 ⊢ ({𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅ ↔ ∩ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ V)
123	121, 122	sylibr 224	. . . . . . 7 ⊢ (Lim 𝐴 → {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅)
124		onint 6995	. . . . . . 7 ⊢ (({𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ⊆ On ∧ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ≠ ∅) → ∩ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
125	119, 123, 124	sylancr 695	. . . . . 6 ⊢ (Lim 𝐴 → ∩ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)})
126	115, 125	sseldi 3601	. . . . 5 ⊢ (Lim 𝐴 → ∩ {𝑥 ∣ ∃𝑦 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ ∪ 𝑦 = 𝐴}𝑥 = (card‘𝑦)} ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
127	108, 126	eqeltrd 2701	. . . 4 ⊢ (Lim 𝐴 → (cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)})
128		sseq2 3627	. . . . . 6 ⊢ (𝑥 = (cf‘𝐴) → (ω ⊆ 𝑥 ↔ ω ⊆ (cf‘𝐴)))
129		limeq 5735	. . . . . 6 ⊢ (𝑥 = (cf‘𝐴) → (Lim 𝑥 ↔ Lim (cf‘𝐴)))
130	128, 129	bibi12d 335	. . . . 5 ⊢ (𝑥 = (cf‘𝐴) → ((ω ⊆ 𝑥 ↔ Lim 𝑥) ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴))))
131	120, 130	elab 3350	. . . 4 ⊢ ((cf‘𝐴) ∈ {𝑥 ∣ (ω ⊆ 𝑥 ↔ Lim 𝑥)} ↔ (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
132	127, 131	sylib 208	. . 3 ⊢ (Lim 𝐴 → (ω ⊆ (cf‘𝐴) ↔ Lim (cf‘𝐴)))
133	105, 132	mpbid 222	. 2 ⊢ (Lim 𝐴 → Lim (cf‘𝐴))
134		eloni 5733	. . . . . . 7 ⊢ (𝐴 ∈ On → Ord 𝐴)
135		ordzsl 7045	. . . . . . 7 ⊢ (Ord 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
136	134, 135	sylib 208	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴))
137		df-3or 1038	. . . . . . 7 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴))
138		orcom 402	. . . . . . 7 ⊢ (((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ∨ Lim 𝐴) ↔ (Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
139		df-or 385	. . . . . . 7 ⊢ ((Lim 𝐴 ∨ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
140	137, 138, 139	3bitri 286	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ Lim 𝐴) ↔ (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
141	136, 140	sylib 208	. . . . 5 ⊢ (𝐴 ∈ On → (¬ Lim 𝐴 → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥)))
142		fveq2 6191	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (cf‘𝐴) = (cf‘∅))
143		cf0 9073	. . . . . . . . 9 ⊢ (cf‘∅) = ∅
144	142, 143	syl6eq 2672	. . . . . . . 8 ⊢ (𝐴 = ∅ → (cf‘𝐴) = ∅)
145		limeq 5735	. . . . . . . 8 ⊢ ((cf‘𝐴) = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
146	144, 145	syl 17	. . . . . . 7 ⊢ (𝐴 = ∅ → (Lim (cf‘𝐴) ↔ Lim ∅))
147	79, 146	mtbiri 317	. . . . . 6 ⊢ (𝐴 = ∅ → ¬ Lim (cf‘𝐴))
148		1n0 7575	. . . . . . . . . 10 ⊢ 1𝑜 ≠ ∅
149		df1o2 7572	. . . . . . . . . . . 12 ⊢ 1𝑜 = {∅}
150	149	unieqi 4445	. . . . . . . . . . 11 ⊢ ∪ 1𝑜 = ∪ {∅}
151		0ex 4790	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
152	151	unisn 4451	. . . . . . . . . . 11 ⊢ ∪ {∅} = ∅
153	150, 152	eqtri 2644	. . . . . . . . . 10 ⊢ ∪ 1𝑜 = ∅
154	148, 153	neeqtrri 2867	. . . . . . . . 9 ⊢ 1𝑜 ≠ ∪ 1𝑜
155		limuni 5785	. . . . . . . . . 10 ⊢ (Lim 1𝑜 → 1𝑜 = ∪ 1𝑜)
156	155	necon3ai 2819	. . . . . . . . 9 ⊢ (1𝑜 ≠ ∪ 1𝑜 → ¬ Lim 1𝑜)
157	154, 156	ax-mp 5	. . . . . . . 8 ⊢ ¬ Lim 1𝑜
158		fveq2 6191	. . . . . . . . . 10 ⊢ (𝐴 = suc 𝑥 → (cf‘𝐴) = (cf‘suc 𝑥))
159		cfsuc 9079	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (cf‘suc 𝑥) = 1𝑜)
160	158, 159	sylan9eqr 2678	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (cf‘𝐴) = 1𝑜)
161		limeq 5735	. . . . . . . . 9 ⊢ ((cf‘𝐴) = 1𝑜 → (Lim (cf‘𝐴) ↔ Lim 1𝑜))
162	160, 161	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → (Lim (cf‘𝐴) ↔ Lim 1𝑜))
163	157, 162	mtbiri 317	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
164	163	rexlimiva 3028	. . . . . 6 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → ¬ Lim (cf‘𝐴))
165	147, 164	jaoi 394	. . . . 5 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥) → ¬ Lim (cf‘𝐴))
166	141, 165	syl6 35	. . . 4 ⊢ (𝐴 ∈ On → (¬ Lim 𝐴 → ¬ Lim (cf‘𝐴)))
167	166	con4d 114	. . 3 ⊢ (𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
168		cff 9070	. . . . . . . . 9 ⊢ cf:On⟶On
169	168	fdmi 6052	. . . . . . . 8 ⊢ dom cf = On
170	169	eleq2i 2693	. . . . . . 7 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
171		ndmfv 6218	. . . . . . 7 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
172	170, 171	sylnbir 321	. . . . . 6 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
173	172, 145	syl 17	. . . . 5 ⊢ (¬ 𝐴 ∈ On → (Lim (cf‘𝐴) ↔ Lim ∅))
174	79, 173	mtbiri 317	. . . 4 ⊢ (¬ 𝐴 ∈ On → ¬ Lim (cf‘𝐴))
175	174	pm2.21d 118	. . 3 ⊢ (¬ 𝐴 ∈ On → (Lim (cf‘𝐴) → Lim 𝐴))
176	167, 175	pm2.61i 176	. 2 ⊢ (Lim (cf‘𝐴) → Lim 𝐴)
177	133, 176	impbii 199	1 ⊢ (Lim 𝐴 ↔ Lim (cf‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ∩ cint 4475  ∩ ciin 4521   class class class wbr 4653   E cep 5028   Or wor 5034   Fr wfr 5070   We wwe 5072  ^◡ccnv 5113  dom cdm 5114  Ord word 5722  Oncon0 5723  Lim wlim 5724  suc csuc 5725  ‘cfv 5888  ωcom 7065  1_𝑜c1o 7553   ≈ cen 7952   ≺ csdm 7954  Fincfn 7955  cardccrd 8761  cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cf 8767

This theorem is referenced by:  cfom  9086