Theorem rankcf 10660

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 9680	. . 3 ⊢ (rank‘𝐴) ∈ On
2		onzsl 7771	. . 3 ⊢ ((rank‘𝐴) ∈ On ↔ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴))))
3	1, 2	mpbi 230	. 2 ⊢ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)))
4		sdom0 9017	. . . 4 ⊢ ¬ 𝐴 ≺ ∅
5		fveq2 6817	. . . . . 6 ⊢ ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = (cf‘∅))
6		cf0 10134	. . . . . 6 ⊢ (cf‘∅) = ∅
7	5, 6	eqtrdi 2781	. . . . 5 ⊢ ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = ∅)
8	7	breq2d 5101	. . . 4 ⊢ ((rank‘𝐴) = ∅ → (𝐴 ≺ (cf‘(rank‘𝐴)) ↔ 𝐴 ≺ ∅))
9	4, 8	mtbiri 327	. . 3 ⊢ ((rank‘𝐴) = ∅ → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
10		fveq2 6817	. . . . . . 7 ⊢ ((rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) = (cf‘suc 𝑥))
11		cfsuc 10140	. . . . . . 7 ⊢ (𝑥 ∈ On → (cf‘suc 𝑥) = 1o)
12	10, 11	sylan9eqr 2787	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) = 1o)
13		nsuceq0 6387	. . . . . . . . 9 ⊢ suc 𝑥 ≠ ∅
14		neeq1 2988	. . . . . . . . 9 ⊢ ((rank‘𝐴) = suc 𝑥 → ((rank‘𝐴) ≠ ∅ ↔ suc 𝑥 ≠ ∅))
15	13, 14	mpbiri 258	. . . . . . . 8 ⊢ ((rank‘𝐴) = suc 𝑥 → (rank‘𝐴) ≠ ∅)
16		fveq2 6817	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅))
17		0elon 6357	. . . . . . . . . . . . 13 ⊢ ∅ ∈ On
18		r1fnon 9652	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
19	18	fndmi 6581	. . . . . . . . . . . . 13 ⊢ dom 𝑅1 = On
20	17, 19	eleqtrri 2828	. . . . . . . . . . . 12 ⊢ ∅ ∈ dom 𝑅1
21		rankonid 9714	. . . . . . . . . . . 12 ⊢ (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅)
22	20, 21	mpbi 230	. . . . . . . . . . 11 ⊢ (rank‘∅) = ∅
23	16, 22	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (rank‘𝐴) = ∅)
24	23	necon3i 2958	. . . . . . . . 9 ⊢ ((rank‘𝐴) ≠ ∅ → 𝐴 ≠ ∅)
25		rankvaln 9684	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∅)
26	25	necon1ai 2953	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ≠ ∅ → 𝐴 ∈ ∪ (𝑅1 “ On))
27		breq2 5093	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (1o ≼ 𝑦 ↔ 1o ≼ 𝐴))
28		neeq1 2988	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
29		0sdom1dom 9125	. . . . . . . . . . . 12 ⊢ (∅ ≺ 𝑦 ↔ 1o ≼ 𝑦)
30		vex 3438	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
31	30	0sdom 9016	. . . . . . . . . . . 12 ⊢ (∅ ≺ 𝑦 ↔ 𝑦 ≠ ∅)
32	29, 31	bitr3i 277	. . . . . . . . . . 11 ⊢ (1o ≼ 𝑦 ↔ 𝑦 ≠ ∅)
33	27, 28, 32	vtoclbg 3510	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (1o ≼ 𝐴 ↔ 𝐴 ≠ ∅))
34	26, 33	syl 17	. . . . . . . . 9 ⊢ ((rank‘𝐴) ≠ ∅ → (1o ≼ 𝐴 ↔ 𝐴 ≠ ∅))
35	24, 34	mpbird 257	. . . . . . . 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 5111	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) ≼ 𝐴)
39	38	rexlimiva 3123	. . . 4 ⊢ (∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) ≼ 𝐴)
40		domnsym 9011	. . . 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 6362	. . . . . . . . . . . . . . . . 17 ⊢ ¬ Lim ∅
43		limeq 6314	. . . . . . . . . . . . . . . . 17 ⊢ ((rank‘𝐴) = ∅ → (Lim (rank‘𝐴) ↔ Lim ∅))
44	42, 43	mtbiri 327	. . . . . . . . . . . . . . . 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 9690	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
48	46, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
49	48	sselda 3932	. . . . . . . . . . . 12 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
50		ranksnb 9712	. . . . . . . . . . . 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 9709	. . . . . . . . . . . . . 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 7774	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ∈ (rank‘𝐴)))
55	53, 54	sylibd 239	. . . . . . . . . . . 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 2829	. . . . . . . . . 10 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (rank‘{𝑥}) ∈ (rank‘𝐴))
58		eleq1a 2824	. . . . . . . . . 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 3131	. . . . . . . 8 ⊢ (Lim (rank‘𝐴) → (∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
61	60	abssdv 4017	. . . . . . 7 ⊢ (Lim (rank‘𝐴) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴))
62		vsnex 5370	. . . . . . . . . . . . 13 ⊢ {𝑥} ∈ V
63	62	dfiun2 4980	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
64		iunid 5007	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
65	63, 64	eqtr3i 2755	. . . . . . . . . . 11 ⊢ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} = 𝐴
66	65	fveq2i 6820	. . . . . . . . . 10 ⊢ (rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = (rank‘𝐴)
67	47	sselda 3932	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
68		snwf 9694	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → {𝑥} ∈ ∪ (𝑅1 “ On))
69		eleq1a 2824	. . . . . . . . . . . . . . 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 3131	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 = {𝑥} → 𝑦 ∈ ∪ (𝑅1 “ On)))
72	71	abssdv 4017	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On))
73		abrexexg 7888	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ V)
74		eleq1 2817	. . . . . . . . . . . . . 14 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On)))
75		sseq1 3958	. . . . . . . . . . . . . 14 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ⊆ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On)))
76		vex 3438	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
77	76	r1elss 9691	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑧 ⊆ ∪ (𝑅1 “ On))
78	74, 75, 77	vtoclbg 3510	. . . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On))
81		rankuni2b 9738	. . . . . . . . . . 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 2779	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧))
84		fvex 6830	. . . . . . . . . . 11 ⊢ (rank‘𝑧) ∈ V
85	84	dfiun2 4980	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)}
86		fveq2 6817	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑥} → (rank‘𝑧) = (rank‘{𝑥}))
87	62, 86	abrexco 7173	. . . . . . . . . . 11 ⊢ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
88	87	unieqi 4869	. . . . . . . . . 10 ⊢ ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
89	85, 88	eqtri 2753	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
90	83, 89	eqtr2di 2782	. . . . . . . 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 6830	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ V
93	92	cfslb 10149	. . . . . . 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 1465	. . . . . 6 ⊢ (Lim (rank‘𝐴) → (cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})})
95		2fveq3 6822	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴)))
96		breq12 5094	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
97	95, 96	mpdan 687	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
98		rexeq 3286	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥}) ↔ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})))
99	98	abbidv 2796	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})})
100		breq12 5094	. . . . . . . . . 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 688	. . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))
104	103	rnmpt 5894	. . . . . . . . 9 ⊢ ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})}
105		cfon 10138	. . . . . . . . . . 11 ⊢ (cf‘(rank‘𝑦)) ∈ On
106		sdomdom 8897	. . . . . . . . . . 11 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ≼ (cf‘(rank‘𝑦)))
107		ondomen 9920	. . . . . . . . . . 11 ⊢ (((cf‘(rank‘𝑦)) ∈ On ∧ 𝑦 ≼ (cf‘(rank‘𝑦))) → 𝑦 ∈ dom card)
108	105, 106, 107	sylancr 587	. . . . . . . . . 10 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ∈ dom card)
109		fvex 6830	. . . . . . . . . . . 12 ⊢ (rank‘{𝑥}) ∈ V
110	109, 103	fnmpti 6620	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦
111		dffn4 6737	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})))
112	110, 111	mpbi 230	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))
113		fodomnum 9940	. . . . . . . . . 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 5125	. . . . . . . 8 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦)
116	102, 115	vtoclg 3507	. . . . . . 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 8924	. . . . . . 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 684	. . . . 5 ⊢ (Lim (rank‘𝐴) → (𝐴 ≺ (cf‘(rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))))
121	120	pm2.01d 190	. . . 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 1430	. 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 206   ∧ wa 395   ∨ w3o 1085   = wceq 1541   ∈ wcel 2110  {cab 2708   ≠ wne 2926  ∃wrex 3054  Vcvv 3434   ⊆ wss 3900  ∅c0 4281  {csn 4574  ∪ cuni 4857  ∪ ciun 4939   class class class wbr 5089   ↦ cmpt 5170  dom cdm 5614  ran crn 5615   “ cima 5617  Oncon0 6302  Lim wlim 6303  suc csuc 6304   Fn wfn 6472  –onto→wfo 6475  ‘cfv 6477  1_oc1o 8373   ≼ cdom 8862   ≺ csdm 8863  𝑅₁cr1 9647  rankcrnk 9648  cardccrd 9820  cfccf 9822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-iin 4942  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-r1 9649  df-rank 9650  df-card 9824  df-cf 9826  df-acn 9827

This theorem is referenced by:  inatsk  10661  grur1  10703