Theorem rankcf 10188

Description: Any set must be at least as large as the cofinality of its rank, because the ranks of the elements of 𝐴 form a cofinal map into (rank‘𝐴). (Contributed by Mario Carneiro, 27-May-2013.)

Assertion

Ref	Expression
rankcf	⊢ ¬ 𝐴 ≺ (cf‘(rank‘𝐴))

Proof of Theorem rankcf

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

Step	Hyp	Ref	Expression
1		rankon 9208	. . 3 ⊢ (rank‘𝐴) ∈ On
2		onzsl 7541	. . 3 ⊢ ((rank‘𝐴) ∈ On ↔ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴))))
3	1, 2	mpbi 233	. 2 ⊢ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)))
4		sdom0 8633	. . . 4 ⊢ ¬ 𝐴 ≺ ∅
5		fveq2 6645	. . . . . 6 ⊢ ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = (cf‘∅))
6		cf0 9662	. . . . . 6 ⊢ (cf‘∅) = ∅
7	5, 6	eqtrdi 2849	. . . . 5 ⊢ ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = ∅)
8	7	breq2d 5042	. . . 4 ⊢ ((rank‘𝐴) = ∅ → (𝐴 ≺ (cf‘(rank‘𝐴)) ↔ 𝐴 ≺ ∅))
9	4, 8	mtbiri 330	. . 3 ⊢ ((rank‘𝐴) = ∅ → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
10		fveq2 6645	. . . . . . 7 ⊢ ((rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) = (cf‘suc 𝑥))
11		cfsuc 9668	. . . . . . 7 ⊢ (𝑥 ∈ On → (cf‘suc 𝑥) = 1o)
12	10, 11	sylan9eqr 2855	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) = 1o)
13		nsuceq0 6239	. . . . . . . . 9 ⊢ suc 𝑥 ≠ ∅
14		neeq1 3049	. . . . . . . . 9 ⊢ ((rank‘𝐴) = suc 𝑥 → ((rank‘𝐴) ≠ ∅ ↔ suc 𝑥 ≠ ∅))
15	13, 14	mpbiri 261	. . . . . . . 8 ⊢ ((rank‘𝐴) = suc 𝑥 → (rank‘𝐴) ≠ ∅)
16		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅))
17		0elon 6212	. . . . . . . . . . . . 13 ⊢ ∅ ∈ On
18		r1fnon 9180	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
19	18	fndmi 6426	. . . . . . . . . . . . 13 ⊢ dom 𝑅1 = On
20	17, 19	eleqtrri 2889	. . . . . . . . . . . 12 ⊢ ∅ ∈ dom 𝑅1
21		rankonid 9242	. . . . . . . . . . . 12 ⊢ (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅)
22	20, 21	mpbi 233	. . . . . . . . . . 11 ⊢ (rank‘∅) = ∅
23	16, 22	eqtrdi 2849	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (rank‘𝐴) = ∅)
24	23	necon3i 3019	. . . . . . . . 9 ⊢ ((rank‘𝐴) ≠ ∅ → 𝐴 ≠ ∅)
25		rankvaln 9212	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∅)
26	25	necon1ai 3014	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ≠ ∅ → 𝐴 ∈ ∪ (𝑅1 “ On))
27		breq2 5034	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (1o ≼ 𝑦 ↔ 1o ≼ 𝐴))
28		neeq1 3049	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
29		0sdom1dom 8700	. . . . . . . . . . . 12 ⊢ (∅ ≺ 𝑦 ↔ 1o ≼ 𝑦)
30		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
31	30	0sdom 8632	. . . . . . . . . . . 12 ⊢ (∅ ≺ 𝑦 ↔ 𝑦 ≠ ∅)
32	29, 31	bitr3i 280	. . . . . . . . . . 11 ⊢ (1o ≼ 𝑦 ↔ 𝑦 ≠ ∅)
33	27, 28, 32	vtoclbg 3517	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (1o ≼ 𝐴 ↔ 𝐴 ≠ ∅))
34	26, 33	syl 17	. . . . . . . . 9 ⊢ ((rank‘𝐴) ≠ ∅ → (1o ≼ 𝐴 ↔ 𝐴 ≠ ∅))
35	24, 34	mpbird 260	. . . . . . . 8 ⊢ ((rank‘𝐴) ≠ ∅ → 1o ≼ 𝐴)
36	15, 35	syl 17	. . . . . . 7 ⊢ ((rank‘𝐴) = suc 𝑥 → 1o ≼ 𝐴)
37	36	adantl 485	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → 1o ≼ 𝐴)
38	12, 37	eqbrtrd 5052	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) ≼ 𝐴)
39	38	rexlimiva 3240	. . . 4 ⊢ (∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) ≼ 𝐴)
40		domnsym 8627	. . . 4 ⊢ ((cf‘(rank‘𝐴)) ≼ 𝐴 → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
41	39, 40	syl 17	. . 3 ⊢ (∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
42		nlim0 6217	. . . . . . . . . . . . . . . . 17 ⊢ ¬ Lim ∅
43		limeq 6171	. . . . . . . . . . . . . . . . 17 ⊢ ((rank‘𝐴) = ∅ → (Lim (rank‘𝐴) ↔ Lim ∅))
44	42, 43	mtbiri 330	. . . . . . . . . . . . . . . 16 ⊢ ((rank‘𝐴) = ∅ → ¬ Lim (rank‘𝐴))
45	25, 44	syl 17	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → ¬ Lim (rank‘𝐴))
46	45	con4i 114	. . . . . . . . . . . . . 14 ⊢ (Lim (rank‘𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
47		r1elssi 9218	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
48	46, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
49	48	sselda 3915	. . . . . . . . . . . 12 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
50		ranksnb 9240	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (rank‘{𝑥}) = suc (rank‘𝑥))
51	49, 50	syl 17	. . . . . . . . . . 11 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (rank‘{𝑥}) = suc (rank‘𝑥))
52		rankelb 9237	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
53	46, 52	syl 17	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
54		limsuc 7544	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ∈ (rank‘𝐴)))
55	53, 54	sylibd 242	. . . . . . . . . . . 12 ⊢ (Lim (rank‘𝐴) → (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ∈ (rank‘𝐴)))
56	55	imp 410	. . . . . . . . . . 11 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → suc (rank‘𝑥) ∈ (rank‘𝐴))
57	51, 56	eqeltrd 2890	. . . . . . . . . 10 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (rank‘{𝑥}) ∈ (rank‘𝐴))
58		eleq1a 2885	. . . . . . . . . 10 ⊢ ((rank‘{𝑥}) ∈ (rank‘𝐴) → (𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
59	57, 58	syl 17	. . . . . . . . 9 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
60	59	rexlimdva 3243	. . . . . . . 8 ⊢ (Lim (rank‘𝐴) → (∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
61	60	abssdv 3996	. . . . . . 7 ⊢ (Lim (rank‘𝐴) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴))
62		snex 5297	. . . . . . . . . . . . 13 ⊢ {𝑥} ∈ V
63	62	dfiun2 4920	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
64		iunid 4947	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
65	63, 64	eqtr3i 2823	. . . . . . . . . . 11 ⊢ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} = 𝐴
66	65	fveq2i 6648	. . . . . . . . . 10 ⊢ (rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = (rank‘𝐴)
67	47	sselda 3915	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
68		snwf 9222	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → {𝑥} ∈ ∪ (𝑅1 “ On))
69		eleq1a 2885	. . . . . . . . . . . . . . 15 ⊢ ({𝑥} ∈ ∪ (𝑅1 “ On) → (𝑦 = {𝑥} → 𝑦 ∈ ∪ (𝑅1 “ On)))
70	67, 68, 69	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → (𝑦 = {𝑥} → 𝑦 ∈ ∪ (𝑅1 “ On)))
71	70	rexlimdva 3243	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 = {𝑥} → 𝑦 ∈ ∪ (𝑅1 “ On)))
72	71	abssdv 3996	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On))
73		abrexexg 7644	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ V)
74		eleq1 2877	. . . . . . . . . . . . . 14 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On)))
75		sseq1 3940	. . . . . . . . . . . . . 14 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ⊆ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On)))
76		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
77	76	r1elss 9219	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑧 ⊆ ∪ (𝑅1 “ On))
78	74, 75, 77	vtoclbg 3517	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ V → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On)))
79	73, 78	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On)))
80	72, 79	mpbird 260	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On))
81		rankuni2b 9266	. . . . . . . . . . 11 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On) → (rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧))
82	80, 81	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧))
83	66, 82	syl5eqr 2847	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧))
84		fvex 6658	. . . . . . . . . . 11 ⊢ (rank‘𝑧) ∈ V
85	84	dfiun2 4920	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)}
86		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑥} → (rank‘𝑧) = (rank‘{𝑥}))
87	62, 86	abrexco 6981	. . . . . . . . . . 11 ⊢ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
88	87	unieqi 4813	. . . . . . . . . 10 ⊢ ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
89	85, 88	eqtri 2821	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
90	83, 89	eqtr2di 2850	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴))
91	46, 90	syl 17	. . . . . . 7 ⊢ (Lim (rank‘𝐴) → ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴))
92		fvex 6658	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ V
93	92	cfslb 9677	. . . . . . 7 ⊢ ((Lim (rank‘𝐴) ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴) ∧ ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴)) → (cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})})
94	61, 91, 93	mpd3an23 1460	. . . . . 6 ⊢ (Lim (rank‘𝐴) → (cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})})
95		2fveq3 6650	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴)))
96		breq12 5035	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
97	95, 96	mpdan 686	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
98		rexeq 3359	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥}) ↔ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})))
99	98	abbidv 2862	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})})
100		breq12 5035	. . . . . . . . . 10 ⊢ (({𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ 𝑦 = 𝐴) → ({𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦 ↔ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
101	99, 100	mpancom 687	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ({𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦 ↔ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
102	97, 101	imbi12d 348	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → ((𝑦 ≺ (cf‘(rank‘𝑦)) → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦) ↔ (𝐴 ≺ (cf‘(rank‘𝐴)) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴)))
103		eqid 2798	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))
104	103	rnmpt 5791	. . . . . . . . 9 ⊢ ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})}
105		cfon 9666	. . . . . . . . . . 11 ⊢ (cf‘(rank‘𝑦)) ∈ On
106		sdomdom 8520	. . . . . . . . . . 11 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ≼ (cf‘(rank‘𝑦)))
107		ondomen 9448	. . . . . . . . . . 11 ⊢ (((cf‘(rank‘𝑦)) ∈ On ∧ 𝑦 ≼ (cf‘(rank‘𝑦))) → 𝑦 ∈ dom card)
108	105, 106, 107	sylancr 590	. . . . . . . . . 10 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ∈ dom card)
109		fvex 6658	. . . . . . . . . . . 12 ⊢ (rank‘{𝑥}) ∈ V
110	109, 103	fnmpti 6463	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦
111		dffn4 6571	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})))
112	110, 111	mpbi 233	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))
113		fodomnum 9468	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) → ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦))
114	108, 112, 113	mpisyl 21	. . . . . . . . 9 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦)
115	104, 114	eqbrtrrid 5066	. . . . . . . 8 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦)
116	102, 115	vtoclg 3515	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 ≺ (cf‘(rank‘𝐴)) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
117	46, 116	syl 17	. . . . . 6 ⊢ (Lim (rank‘𝐴) → (𝐴 ≺ (cf‘(rank‘𝐴)) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
118		domtr 8545	. . . . . . 7 ⊢ (((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → (cf‘(rank‘𝐴)) ≼ 𝐴)
119	118, 40	syl 17	. . . . . 6 ⊢ (((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
120	94, 117, 119	syl6an 683	. . . . 5 ⊢ (Lim (rank‘𝐴) → (𝐴 ≺ (cf‘(rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))))
121	120	pm2.01d 193	. . . 4 ⊢ (Lim (rank‘𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
122	121	adantl 485	. . 3 ⊢ (((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
123	9, 41, 122	3jaoi 1424	. 2 ⊢ (((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴))) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
124	3, 123	ax-mp 5	1 ⊢ ¬ 𝐴 ≺ (cf‘(rank‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  {csn 4525  ∪ cuni 4800  ∪ ciun 4881   class class class wbr 5030   ↦ cmpt 5110  dom cdm 5519  ran crn 5520   “ cima 5522  Oncon0 6159  Lim wlim 6160  suc csuc 6161   Fn wfn 6319  –onto→wfo 6322  ‘cfv 6324  1_oc1o 8078   ≼ cdom 8490   ≺ csdm 8491  𝑅₁cr1 9175  rankcrnk 9176  cardccrd 9348  cfccf 9350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-r1 9177  df-rank 9178  df-card 9352  df-cf 9354  df-acn 9355

This theorem is referenced by:  inatsk  10189  grur1  10231