Theorem rankcf 9599

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 8658	. . 3 ⊢ (rank‘𝐴) ∈ On
2		onzsl 7046	. . 3 ⊢ ((rank‘𝐴) ∈ On ↔ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴))))
3	1, 2	mpbi 220	. 2 ⊢ ((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)))
4		sdom0 8092	. . . 4 ⊢ ¬ 𝐴 ≺ ∅
5		fveq2 6191	. . . . . 6 ⊢ ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = (cf‘∅))
6		cf0 9073	. . . . . 6 ⊢ (cf‘∅) = ∅
7	5, 6	syl6eq 2672	. . . . 5 ⊢ ((rank‘𝐴) = ∅ → (cf‘(rank‘𝐴)) = ∅)
8	7	breq2d 4665	. . . 4 ⊢ ((rank‘𝐴) = ∅ → (𝐴 ≺ (cf‘(rank‘𝐴)) ↔ 𝐴 ≺ ∅))
9	4, 8	mtbiri 317	. . 3 ⊢ ((rank‘𝐴) = ∅ → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
10		fveq2 6191	. . . . . . 7 ⊢ ((rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) = (cf‘suc 𝑥))
11		cfsuc 9079	. . . . . . 7 ⊢ (𝑥 ∈ On → (cf‘suc 𝑥) = 1𝑜)
12	10, 11	sylan9eqr 2678	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) = 1𝑜)
13		nsuceq0 5805	. . . . . . . . 9 ⊢ suc 𝑥 ≠ ∅
14		neeq1 2856	. . . . . . . . 9 ⊢ ((rank‘𝐴) = suc 𝑥 → ((rank‘𝐴) ≠ ∅ ↔ suc 𝑥 ≠ ∅))
15	13, 14	mpbiri 248	. . . . . . . 8 ⊢ ((rank‘𝐴) = suc 𝑥 → (rank‘𝐴) ≠ ∅)
16		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (rank‘𝐴) = (rank‘∅))
17		0elon 5778	. . . . . . . . . . . . 13 ⊢ ∅ ∈ On
18		r1fnon 8630	. . . . . . . . . . . . . 14 ⊢ 𝑅1 Fn On
19		fndm 5990	. . . . . . . . . . . . . 14 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
20	18, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom 𝑅1 = On
21	17, 20	eleqtrri 2700	. . . . . . . . . . . 12 ⊢ ∅ ∈ dom 𝑅1
22		rankonid 8692	. . . . . . . . . . . 12 ⊢ (∅ ∈ dom 𝑅1 ↔ (rank‘∅) = ∅)
23	21, 22	mpbi 220	. . . . . . . . . . 11 ⊢ (rank‘∅) = ∅
24	16, 23	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (rank‘𝐴) = ∅)
25	24	necon3i 2826	. . . . . . . . 9 ⊢ ((rank‘𝐴) ≠ ∅ → 𝐴 ≠ ∅)
26		rankvaln 8662	. . . . . . . . . . 11 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∅)
27	26	necon1ai 2821	. . . . . . . . . 10 ⊢ ((rank‘𝐴) ≠ ∅ → 𝐴 ∈ ∪ (𝑅1 “ On))
28		breq2 4657	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (1𝑜 ≼ 𝑦 ↔ 1𝑜 ≼ 𝐴))
29		neeq1 2856	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑦 ≠ ∅ ↔ 𝐴 ≠ ∅))
30		0sdom1dom 8158	. . . . . . . . . . . 12 ⊢ (∅ ≺ 𝑦 ↔ 1𝑜 ≼ 𝑦)
31		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
32	31	0sdom 8091	. . . . . . . . . . . 12 ⊢ (∅ ≺ 𝑦 ↔ 𝑦 ≠ ∅)
33	30, 32	bitr3i 266	. . . . . . . . . . 11 ⊢ (1𝑜 ≼ 𝑦 ↔ 𝑦 ≠ ∅)
34	28, 29, 33	vtoclbg 3267	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (1𝑜 ≼ 𝐴 ↔ 𝐴 ≠ ∅))
35	27, 34	syl 17	. . . . . . . . 9 ⊢ ((rank‘𝐴) ≠ ∅ → (1𝑜 ≼ 𝐴 ↔ 𝐴 ≠ ∅))
36	25, 35	mpbird 247	. . . . . . . 8 ⊢ ((rank‘𝐴) ≠ ∅ → 1𝑜 ≼ 𝐴)
37	15, 36	syl 17	. . . . . . 7 ⊢ ((rank‘𝐴) = suc 𝑥 → 1𝑜 ≼ 𝐴)
38	37	adantl 482	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → 1𝑜 ≼ 𝐴)
39	12, 38	eqbrtrd 4675	. . . . 5 ⊢ ((𝑥 ∈ On ∧ (rank‘𝐴) = suc 𝑥) → (cf‘(rank‘𝐴)) ≼ 𝐴)
40	39	rexlimiva 3028	. . . 4 ⊢ (∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 → (cf‘(rank‘𝐴)) ≼ 𝐴)
41		domnsym 8086	. . . 4 ⊢ ((cf‘(rank‘𝐴)) ≼ 𝐴 → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
42	40, 41	syl 17	. . 3 ⊢ (∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
43		nlim0 5783	. . . . . . . . . . . . . . . . 17 ⊢ ¬ Lim ∅
44		limeq 5735	. . . . . . . . . . . . . . . . 17 ⊢ ((rank‘𝐴) = ∅ → (Lim (rank‘𝐴) ↔ Lim ∅))
45	43, 44	mtbiri 317	. . . . . . . . . . . . . . . 16 ⊢ ((rank‘𝐴) = ∅ → ¬ Lim (rank‘𝐴))
46	26, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐴 ∈ ∪ (𝑅1 “ On) → ¬ Lim (rank‘𝐴))
47	46	con4i 113	. . . . . . . . . . . . . 14 ⊢ (Lim (rank‘𝐴) → 𝐴 ∈ ∪ (𝑅1 “ On))
48		r1elssi 8668	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ⊆ ∪ (𝑅1 “ On))
49	47, 48	syl 17	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
50	49	sselda 3603	. . . . . . . . . . . 12 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
51		ranksnb 8690	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (rank‘{𝑥}) = suc (rank‘𝑥))
52	50, 51	syl 17	. . . . . . . . . . 11 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (rank‘{𝑥}) = suc (rank‘𝑥))
53		rankelb 8687	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
54	47, 53	syl 17	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → (𝑥 ∈ 𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
55		limsuc 7049	. . . . . . . . . . . . 13 ⊢ (Lim (rank‘𝐴) → ((rank‘𝑥) ∈ (rank‘𝐴) ↔ suc (rank‘𝑥) ∈ (rank‘𝐴)))
56	54, 55	sylibd 229	. . . . . . . . . . . 12 ⊢ (Lim (rank‘𝐴) → (𝑥 ∈ 𝐴 → suc (rank‘𝑥) ∈ (rank‘𝐴)))
57	56	imp 445	. . . . . . . . . . 11 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → suc (rank‘𝑥) ∈ (rank‘𝐴))
58	52, 57	eqeltrd 2701	. . . . . . . . . 10 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (rank‘{𝑥}) ∈ (rank‘𝐴))
59		eleq1a 2696	. . . . . . . . . 10 ⊢ ((rank‘{𝑥}) ∈ (rank‘𝐴) → (𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
60	58, 59	syl 17	. . . . . . . . 9 ⊢ ((Lim (rank‘𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
61	60	rexlimdva 3031	. . . . . . . 8 ⊢ (Lim (rank‘𝐴) → (∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥}) → 𝑤 ∈ (rank‘𝐴)))
62	61	abssdv 3676	. . . . . . 7 ⊢ (Lim (rank‘𝐴) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ⊆ (rank‘𝐴))
63		snex 4908	. . . . . . . . . . . . 13 ⊢ {𝑥} ∈ V
64	63	dfiun2 4554	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}
65		iunid 4575	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴
66	64, 65	eqtr3i 2646	. . . . . . . . . . 11 ⊢ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} = 𝐴
67	66	fveq2i 6194	. . . . . . . . . 10 ⊢ (rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = (rank‘𝐴)
68	48	sselda 3603	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
69		snwf 8672	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → {𝑥} ∈ ∪ (𝑅1 “ On))
70		eleq1a 2696	. . . . . . . . . . . . . . 15 ⊢ ({𝑥} ∈ ∪ (𝑅1 “ On) → (𝑦 = {𝑥} → 𝑦 ∈ ∪ (𝑅1 “ On)))
71	68, 69, 70	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝑥 ∈ 𝐴) → (𝑦 = {𝑥} → 𝑦 ∈ ∪ (𝑅1 “ On)))
72	71	rexlimdva 3031	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (∃𝑥 ∈ 𝐴 𝑦 = {𝑥} → 𝑦 ∈ ∪ (𝑅1 “ On)))
73	72	abssdv 3676	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On))
74		abrexexg 7140	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ V)
75		eleq1 2689	. . . . . . . . . . . . . 14 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On)))
76		sseq1 3626	. . . . . . . . . . . . . 14 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} → (𝑧 ⊆ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On)))
77		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
78	77	r1elss 8669	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ (𝑅1 “ On) ↔ 𝑧 ⊆ ∪ (𝑅1 “ On))
79	75, 76, 78	vtoclbg 3267	. . . . . . . . . . . . 13 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ V → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On)))
80	74, 79	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On) ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ⊆ ∪ (𝑅1 “ On)))
81	73, 80	mpbird 247	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On))
82		rankuni2b 8716	. . . . . . . . . . 11 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} ∈ ∪ (𝑅1 “ On) → (rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧))
83	81, 82	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧))
84	67, 83	syl5eqr 2670	. . . . . . . . 9 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (rank‘𝐴) = ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧))
85		fvex 6201	. . . . . . . . . . 11 ⊢ (rank‘𝑧) ∈ V
86	85	dfiun2 4554	. . . . . . . . . 10 ⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)}
87		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑥} → (rank‘𝑧) = (rank‘{𝑥}))
88	63, 87	abrexco 6502	. . . . . . . . . . 11 ⊢ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
89	88	unieqi 4445	. . . . . . . . . 10 ⊢ ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}}𝑤 = (rank‘𝑧)} = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
90	86, 89	eqtri 2644	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = {𝑥}} (rank‘𝑧) = ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})}
91	84, 90	syl6req 2673	. . . . . . . 8 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴))
92	47, 91	syl 17	. . . . . . 7 ⊢ (Lim (rank‘𝐴) → ∪ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} = (rank‘𝐴))
93		fvex 6201	. . . . . . . 8 ⊢ (rank‘𝐴) ∈ V
94	93	cfslb 9088	. . . . . . 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})})
95	62, 92, 94	mpd3an23 1426	. . . . . 6 ⊢ (Lim (rank‘𝐴) → (cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})})
96		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (rank‘𝑦) = (rank‘𝐴))
97	96	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴)))
98		breq12 4658	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ (cf‘(rank‘𝑦)) = (cf‘(rank‘𝐴))) → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
99	97, 98	mpdan 702	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑦 ≺ (cf‘(rank‘𝑦)) ↔ 𝐴 ≺ (cf‘(rank‘𝐴))))
100		rexeq 3139	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥}) ↔ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})))
101	100	abbidv 2741	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})})
102		breq12 4658	. . . . . . . . . 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))
103	101, 102	mpancom 703	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ({𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦 ↔ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
104	99, 103	imbi12d 334	. . . . . . . 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)))
105		eqid 2622	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))
106	105	rnmpt 5371	. . . . . . . . 9 ⊢ ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) = {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})}
107		cfon 9077	. . . . . . . . . . 11 ⊢ (cf‘(rank‘𝑦)) ∈ On
108		sdomdom 7983	. . . . . . . . . . 11 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ≼ (cf‘(rank‘𝑦)))
109		ondomen 8860	. . . . . . . . . . 11 ⊢ (((cf‘(rank‘𝑦)) ∈ On ∧ 𝑦 ≼ (cf‘(rank‘𝑦))) → 𝑦 ∈ dom card)
110	107, 108, 109	sylancr 695	. . . . . . . . . 10 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → 𝑦 ∈ dom card)
111		fvex 6201	. . . . . . . . . . . 12 ⊢ (rank‘{𝑥}) ∈ V
112	111, 105	fnmpti 6022	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦
113		dffn4 6121	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})))
114	112, 113	mpbi 220	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥}))
115		fodomnum 8880	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom card → ((𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})):𝑦–onto→ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) → ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦))
116	110, 114, 115	mpisyl 21	. . . . . . . . 9 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → ran (𝑥 ∈ 𝑦 ↦ (rank‘{𝑥})) ≼ 𝑦)
117	106, 116	syl5eqbrr 4689	. . . . . . . 8 ⊢ (𝑦 ≺ (cf‘(rank‘𝑦)) → {𝑤 ∣ ∃𝑥 ∈ 𝑦 𝑤 = (rank‘{𝑥})} ≼ 𝑦)
118	104, 117	vtoclg 3266	. . . . . . 7 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → (𝐴 ≺ (cf‘(rank‘𝐴)) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
119	47, 118	syl 17	. . . . . 6 ⊢ (Lim (rank‘𝐴) → (𝐴 ≺ (cf‘(rank‘𝐴)) → {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴))
120		domtr 8009	. . . . . . 7 ⊢ (((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → (cf‘(rank‘𝐴)) ≼ 𝐴)
121	120, 41	syl 17	. . . . . 6 ⊢ (((cf‘(rank‘𝐴)) ≼ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ∧ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 = (rank‘{𝑥})} ≼ 𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
122	95, 119, 121	syl6an 568	. . . . 5 ⊢ (Lim (rank‘𝐴) → (𝐴 ≺ (cf‘(rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴))))
123	122	pm2.01d 181	. . . 4 ⊢ (Lim (rank‘𝐴) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
124	123	adantl 482	. . 3 ⊢ (((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴)) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
125	9, 42, 124	3jaoi 1391	. 2 ⊢ (((rank‘𝐴) = ∅ ∨ ∃𝑥 ∈ On (rank‘𝐴) = suc 𝑥 ∨ ((rank‘𝐴) ∈ V ∧ Lim (rank‘𝐴))) → ¬ 𝐴 ≺ (cf‘(rank‘𝐴)))
126	3, 125	ax-mp 5	1 ⊢ ¬ 𝐴 ≺ (cf‘(rank‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∨ w3o 1036   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ran crn 5115   “ cima 5117  Oncon0 5723  Lim wlim 5724  suc csuc 5725   Fn wfn 5883  –onto→wfo 5886  ‘cfv 5888  1_𝑜c1o 7553   ≼ cdom 7953   ≺ csdm 7954  𝑅₁cr1 8625  rankcrnk 8626  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

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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-r1 8627  df-rank 8628  df-card 8765  df-cf 8767  df-acn 8768

This theorem is referenced by:  inatsk  9600  grur1  9642