Theorem rankcf 10464

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 9484	. . 3 ⊢ (rank‘𝐴) ∈ On
2		onzsl 7668	. . 3 ⊢ ((rank‘𝐴) ∈ On ↔ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴))))
3	1, 2	mpbi 229	. 2 ⊢ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)))
4		sdom0 8845	. . . 4 ⊢ ¬ 𝐴 ≺ ∅
5		fveq2 6756	. . . . . 6 ⊢ ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = (cf‘∅))
6		cf0 9938	. . . . . 6 ⊢ (cf‘∅) = ∅
7	5, 6	eqtrdi 2795	. . . . 5 ⊢ ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = ∅)
8	7	breq2d 5082	. . . 4 ⊢ ((rank‘𝐴) = ∅ → (𝐴 ≺ (cf‘(rank‘𝐴)) ↔ 𝐴 ≺ ∅))
9	4, 8	mtbiri 326	. . 3 ⊢ ((rank‘𝐴) = ∅ → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
10		fveq2 6756	. . . . . . 7 ⊢ ((rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) = (cf‘suc 𝑥))
11		cfsuc 9944	. . . . . . 7 ⊢ (𝑥 ∈ On → (cf‘suc 𝑥) = 1o)
12	10, 11	sylan9eqr 2801	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) = 1o)
13		nsuceq0 6331	. . . . . . . . 9 ⊢ suc 𝑥 ≠ ∅
14		neeq1 3005	. . . . . . . . 9 ⊢ ((rank‘𝐴) = suc 𝑥 → ((rank‘𝐴) ≠ ∅ ↔ suc 𝑥 ≠ ∅))
15	13, 14	mpbiri 257	. . . . . . . 8 ⊢ ((rank‘𝐴) = suc 𝑥 → (rank‘𝐴) ≠ ∅)
16		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅))
17		0elon 6304	. . . . . . . . . . . . 13 ⊢ ∅ ∈ On
18		r1fnon 9456	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
19	18	fndmi 6521	. . . . . . . . . . . . 13 ⊢ dom 𝑅1 = On
20	17, 19	eleqtrri 2838	. . . . . . . . . . . 12 ⊢ ∅ ∈ dom 𝑅1
21		rankonid 9518	. . . . . . . . . . . 12 ⊢ (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅)
22	20, 21	mpbi 229	. . . . . . . . . . 11 ⊢ (rank‘∅) = ∅
23	16, 22	eqtrdi 2795	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (rank‘𝐴) = ∅)
24	23	necon3i 2975	. . . . . . . . 9 ⊢ ((rank‘𝐴) ≠ ∅ → 𝐴 ≠ ∅)
25		rankvaln 9488	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∅)
26	25	necon1ai 2970	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ≠ ∅ → 𝐴 ∈ ∪ (𝑅1 “ On))
27		breq2 5074	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (1o ≼ 𝑦 ↔ 1o ≼ 𝐴))
28		neeq1 3005	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
29		0sdom1dom 8950	. . . . . . . . . . . 12 ⊢ (∅ ≺ 𝑦 ↔ 1o ≼ 𝑦)
30		vex 3426	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
31	30	0sdom 8844	. . . . . . . . . . . 12 ⊢ (∅ ≺ 𝑦 ↔ 𝑦 ≠ ∅)
32	29, 31	bitr3i 276	. . . . . . . . . . 11 ⊢ (1o ≼ 𝑦 ↔ 𝑦 ≠ ∅)
33	27, 28, 32	vtoclbg 3497	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (1o ≼ 𝐴 ↔ 𝐴 ≠ ∅))
34	26, 33	syl 17	. . . . . . . . 9 ⊢ ((rank‘𝐴) ≠ ∅ → (1o ≼ 𝐴 ↔ 𝐴 ≠ ∅))
35	24, 34	mpbird 256	. . . . . . . 8 ⊢ ((rank‘𝐴) ≠ ∅ → 1o ≼ 𝐴)
36	15, 35	syl 17	. . . . . . 7 ⊢ ((rank‘𝐴) = suc 𝑥 → 1o ≼ 𝐴)
37	36	adantl 481	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → 1o ≼ 𝐴)
38	12, 37	eqbrtrd 5092	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) ≼ 𝐴)
39	38	rexlimiva 3209	. . . 4 ⊢ (∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) ≼ 𝐴)
40		domnsym 8839	. . . 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 6309	. . . . . . . . . . . . . . . . 17 ⊢ ¬ Lim ∅
43		limeq 6263	. . . . . . . . . . . . . . . . 17 ⊢ ((rank‘𝐴) = ∅ → (Lim (rank‘𝐴) ↔ Lim ∅))
44	42, 43	mtbiri 326	. . . . . . . . . . . . . . . 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 9494	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
48	46, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
49	48	sselda 3917	. . . . . . . . . . . 12 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
50		ranksnb 9516	. . . . . . . . . . . 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 9513	. . . . . . . . . . . . . 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 7671	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ∈ (rank‘𝐴)))
55	53, 54	sylibd 238	. . . . . . . . . . . 12 ⊢ (Lim (rank‘𝐴) → (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ∈ (rank‘𝐴)))
56	55	imp 406	. . . . . . . . . . 11 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → suc (rank‘𝑥) ∈ (rank‘𝐴))
57	51, 56	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (rank‘{𝑥}) ∈ (rank‘𝐴))
58		eleq1a 2834	. . . . . . . . . 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 3212	. . . . . . . 8 ⊢ (Lim (rank‘𝐴) → (∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
61	60	abssdv 3998	. . . . . . 7 ⊢ (Lim (rank‘𝐴) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴))
62		snex 5349	. . . . . . . . . . . . 13 ⊢ {𝑥} ∈ V
63	62	dfiun2 4959	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
64		iunid 4986	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
65	63, 64	eqtr3i 2768	. . . . . . . . . . 11 ⊢ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} = 𝐴
66	65	fveq2i 6759	. . . . . . . . . 10 ⊢ (rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = (rank‘𝐴)
67	47	sselda 3917	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
68		snwf 9498	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → {𝑥} ∈ ∪ (𝑅1 “ On))
69		eleq1a 2834	. . . . . . . . . . . . . . 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 3212	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 = {𝑥} → 𝑦 ∈ ∪ (𝑅1 “ On)))
72	71	abssdv 3998	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On))
73		abrexexg 7777	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ V)
74		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On)))
75		sseq1 3942	. . . . . . . . . . . . . 14 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ⊆ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On)))
76		vex 3426	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
77	76	r1elss 9495	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑧 ⊆ ∪ (𝑅1 “ On))
78	74, 75, 77	vtoclbg 3497	. . . . . . . . . . . . 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 256	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On))
81		rankuni2b 9542	. . . . . . . . . . 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	eqtr3id 2793	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧))
84		fvex 6769	. . . . . . . . . . 11 ⊢ (rank‘𝑧) ∈ V
85	84	dfiun2 4959	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)}
86		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑥} → (rank‘𝑧) = (rank‘{𝑥}))
87	62, 86	abrexco 7099	. . . . . . . . . . 11 ⊢ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
88	87	unieqi 4849	. . . . . . . . . 10 ⊢ ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
89	85, 88	eqtri 2766	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
90	83, 89	eqtr2di 2796	. . . . . . . 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 6769	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ V
93	92	cfslb 9953	. . . . . . 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 1461	. . . . . 6 ⊢ (Lim (rank‘𝐴) → (cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})})
95		2fveq3 6761	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴)))
96		breq12 5075	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
97	95, 96	mpdan 683	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
98		rexeq 3334	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥}) ↔ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})))
99	98	abbidv 2808	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})})
100		breq12 5075	. . . . . . . . . 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 684	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ({𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦 ↔ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
102	97, 101	imbi12d 344	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))
104	103	rnmpt 5853	. . . . . . . . 9 ⊢ ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})}
105		cfon 9942	. . . . . . . . . . 11 ⊢ (cf‘(rank‘𝑦)) ∈ On
106		sdomdom 8723	. . . . . . . . . . 11 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ≼ (cf‘(rank‘𝑦)))
107		ondomen 9724	. . . . . . . . . . 11 ⊢ (((cf‘(rank‘𝑦)) ∈ On ∧ 𝑦 ≼ (cf‘(rank‘𝑦))) → 𝑦 ∈ dom card)
108	105, 106, 107	sylancr 586	. . . . . . . . . 10 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ∈ dom card)
109		fvex 6769	. . . . . . . . . . . 12 ⊢ (rank‘{𝑥}) ∈ V
110	109, 103	fnmpti 6560	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦
111		dffn4 6678	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})))
112	110, 111	mpbi 229	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))
113		fodomnum 9744	. . . . . . . . . 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 5106	. . . . . . . 8 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦)
116	102, 115	vtoclg 3495	. . . . . . 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 8748	. . . . . . 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 680	. . . . 5 ⊢ (Lim (rank‘𝐴) → (𝐴 ≺ (cf‘(rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))))
121	120	pm2.01d 189	. . . 4 ⊢ (Lim (rank‘𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
122	121	adantl 481	. . 3 ⊢ (((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
123	9, 41, 122	3jaoi 1425	. 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 205   ∧ wa 395   ∨ w3o 1084   = wceq 1539   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ cuni 4836  ∪ ciun 4921   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  ran crn 5581   “ cima 5583  Oncon0 6251  Lim wlim 6252  suc csuc 6253   Fn wfn 6413  –onto→wfo 6416  ‘cfv 6418  1_oc1o 8260   ≼ cdom 8689   ≺ csdm 8690  𝑅₁cr1 9451  rankcrnk 9452  cardccrd 9624  cfccf 9626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-r1 9453  df-rank 9454  df-card 9628  df-cf 9630  df-acn 9631

This theorem is referenced by:  inatsk  10465  grur1  10507