Theorem nchoicelem19 6307

Description: Lemma for nchoice 6308. Assuming well-ordering, there is a cardinal with a finite special set that is its own T-raising. Theorem 7.3 of [Specker] p. 974. (Contributed by SF, 20-Mar-2015.)

Assertion

Ref	Expression
nchoicelem19	⊢ ( ≤c We NC → ∃m ∈ NC (( Spac ‘m) ∈ Fin ∧ Tc m = m))

Proof of Theorem nchoicelem19

Dummy variables n x p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nchoicelem18 6306	. . 3 ⊢ {x ∣ ( Spac ‘x) ∈ Fin } ∈ V
2		fveq2 5328	. . . 4 ⊢ (x = m → ( Spac ‘x) = ( Spac ‘m))
3	2	eleq1d 2419	. . 3 ⊢ (x = m → (( Spac ‘x) ∈ Fin ↔ ( Spac ‘m) ∈ Fin ))
4		fveq2 5328	. . . 4 ⊢ (x = n → ( Spac ‘x) = ( Spac ‘n))
5	4	eleq1d 2419	. . 3 ⊢ (x = n → (( Spac ‘x) ∈ Fin ↔ ( Spac ‘n) ∈ Fin ))
6		id 19	. . 3 ⊢ ( ≤c We NC → ≤c We NC )
7		vvex 4109	. . . . 5 ⊢ V ∈ V
8	7	ncelncsi 6121	. . . 4 ⊢ Nc V ∈ NC
9		ltcpw1pwg 6202	. . . . . . . . 9 ⊢ (V ∈ V → Nc ℘1V <c Nc ℘V)
10	7, 9	ax-mp 5	. . . . . . . 8 ⊢ Nc ℘1V <c Nc ℘V
11		df1c2 4168	. . . . . . . . 9 ⊢ 1c = ℘1V
12	11	nceqi 6109	. . . . . . . 8 ⊢ Nc 1c = Nc ℘1V
13		pwv 3886	. . . . . . . . . 10 ⊢ ℘V = V
14	13	nceqi 6109	. . . . . . . . 9 ⊢ Nc ℘V = Nc V
15	14	eqcomi 2357	. . . . . . . 8 ⊢ Nc V = Nc ℘V
16	10, 12, 15	3brtr4i 4667	. . . . . . 7 ⊢ Nc 1c <c Nc V
17		nchoicelem8 6296	. . . . . . . 8 ⊢ (( ≤c We NC ∧ Nc V ∈ NC ) → (¬ ( Nc V ↑c 0c) ∈ NC ↔ Nc 1c <c Nc V))
18	8, 17	mpan2 652	. . . . . . 7 ⊢ ( ≤c We NC → (¬ ( Nc V ↑c 0c) ∈ NC ↔ Nc 1c <c Nc V))
19	16, 18	mpbiri 224	. . . . . 6 ⊢ ( ≤c We NC → ¬ ( Nc V ↑c 0c) ∈ NC )
20		nchoicelem3 6291	. . . . . 6 ⊢ (( Nc V ∈ NC ∧ ¬ ( Nc V ↑c 0c) ∈ NC ) → ( Spac ‘ Nc V) = { Nc V})
21	8, 19, 20	sylancr 644	. . . . 5 ⊢ ( ≤c We NC → ( Spac ‘ Nc V) = { Nc V})
22		snfi 4431	. . . . 5 ⊢ { Nc V} ∈ Fin
23	21, 22	syl6eqel 2441	. . . 4 ⊢ ( ≤c We NC → ( Spac ‘ Nc V) ∈ Fin )
24		fveq2 5328	. . . . . 6 ⊢ (x = Nc V → ( Spac ‘x) = ( Spac ‘ Nc V))
25	24	eleq1d 2419	. . . . 5 ⊢ (x = Nc V → (( Spac ‘x) ∈ Fin ↔ ( Spac ‘ Nc V) ∈ Fin ))
26	25	rspcev 2955	. . . 4 ⊢ (( Nc V ∈ NC ∧ ( Spac ‘ Nc V) ∈ Fin ) → ∃x ∈ NC ( Spac ‘x) ∈ Fin )
27	8, 23, 26	sylancr 644	. . 3 ⊢ ( ≤c We NC → ∃x ∈ NC ( Spac ‘x) ∈ Fin )
28	1, 3, 5, 6, 27	weds 5938	. 2 ⊢ ( ≤c We NC → ∃m ∈ NC (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n)))
29		simpll 730	. . . . . . 7 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → ≤c We NC )
30		df-we 5906	. . . . . . . . . . 11 ⊢ We = ( Or ∩ Fr )
31	30	breqi 4645	. . . . . . . . . 10 ⊢ ( ≤c We NC ↔ ≤c ( Or ∩ Fr ) NC )
32		brin 4693	. . . . . . . . . 10 ⊢ ( ≤c ( Or ∩ Fr ) NC ↔ ( ≤c Or NC ∧ ≤c Fr NC ))
33	31, 32	bitri 240	. . . . . . . . 9 ⊢ ( ≤c We NC ↔ ( ≤c Or NC ∧ ≤c Fr NC ))
34	33	simplbi 446	. . . . . . . 8 ⊢ ( ≤c We NC → ≤c Or NC )
35		sopc 5934	. . . . . . . . . 10 ⊢ ( ≤c Or NC ↔ ( ≤c Po NC ∧ ≤c Connex NC ))
36	35	simplbi 446	. . . . . . . . 9 ⊢ ( ≤c Or NC → ≤c Po NC )
37		porta 5933	. . . . . . . . . 10 ⊢ ( ≤c Po NC ↔ ( ≤c Ref NC ∧ ≤c Trans NC ∧ ≤c Antisym NC ))
38	37	simp3bi 972	. . . . . . . . 9 ⊢ ( ≤c Po NC → ≤c Antisym NC )
39	36, 38	syl 15	. . . . . . . 8 ⊢ ( ≤c Or NC → ≤c Antisym NC )
40	34, 39	syl 15	. . . . . . 7 ⊢ ( ≤c We NC → ≤c Antisym NC )
41	29, 40	syl 15	. . . . . 6 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → ≤c Antisym NC )
42		simplr 731	. . . . . . 7 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → m ∈ NC )
43		tccl 6160	. . . . . . 7 ⊢ (m ∈ NC → Tc m ∈ NC )
44	42, 43	syl 15	. . . . . 6 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → Tc m ∈ NC )
45		simprr 733	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))
46		simprl 732	. . . . . . . . . 10 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → ( Spac ‘m) ∈ Fin )
47		nchoicelem17 6305	. . . . . . . . . 10 ⊢ (( ≤c We NC ∧ m ∈ NC ∧ ( Spac ‘m) ∈ Fin ) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))
48	29, 42, 46, 47	syl3anc 1182	. . . . . . . . 9 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → (( Spac ‘ Tc m) ∈ Fin ∧ ( Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 1c) ∨ Nc ( Spac ‘ Tc m) = ( Tc Nc ( Spac ‘m) +c 2c))))
49	48	simpld 445	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → ( Spac ‘ Tc m) ∈ Fin )
50		fveq2 5328	. . . . . . . . . . 11 ⊢ (n = Tc m → ( Spac ‘n) = ( Spac ‘ Tc m))
51	50	eleq1d 2419	. . . . . . . . . 10 ⊢ (n = Tc m → (( Spac ‘n) ∈ Fin ↔ ( Spac ‘ Tc m) ∈ Fin ))
52		breq2 4643	. . . . . . . . . 10 ⊢ (n = Tc m → (m ≤c n ↔ m ≤c Tc m))
53	51, 52	imbi12d 311	. . . . . . . . 9 ⊢ (n = Tc m → ((( Spac ‘n) ∈ Fin → m ≤c n) ↔ (( Spac ‘ Tc m) ∈ Fin → m ≤c Tc m)))
54	53	rspcv 2951	. . . . . . . 8 ⊢ ( Tc m ∈ NC → (∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n) → (( Spac ‘ Tc m) ∈ Fin → m ≤c Tc m)))
55	44, 45, 49, 54	syl3c 57	. . . . . . 7 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → m ≤c Tc m)
56		letc 6231	. . . . . . . . . 10 ⊢ ((m ∈ NC ∧ m ∈ NC ∧ m ≤c Tc m) → ∃p ∈ NC m = Tc p)
57	56	3expia 1153	. . . . . . . . 9 ⊢ ((m ∈ NC ∧ m ∈ NC ) → (m ≤c Tc m → ∃p ∈ NC m = Tc p))
58	42, 42, 57	syl2anc 642	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → (m ≤c Tc m → ∃p ∈ NC m = Tc p))
59		nchoicelem12 6300	. . . . . . . . . . . . . . . 16 ⊢ ((p ∈ NC ∧ ( Spac ‘ Tc p) ∈ Fin ) → ( Spac ‘p) ∈ Fin )
60	59	ad2ant2lr 728	. . . . . . . . . . . . . . 15 ⊢ ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n))) → ( Spac ‘p) ∈ Fin )
61		fveq2 5328	. . . . . . . . . . . . . . . . . . . 20 ⊢ (n = p → ( Spac ‘n) = ( Spac ‘p))
62	61	eleq1d 2419	. . . . . . . . . . . . . . . . . . 19 ⊢ (n = p → (( Spac ‘n) ∈ Fin ↔ ( Spac ‘p) ∈ Fin ))
63		breq2 4643	. . . . . . . . . . . . . . . . . . 19 ⊢ (n = p → ( Tc p ≤c n ↔ Tc p ≤c p))
64	62, 63	imbi12d 311	. . . . . . . . . . . . . . . . . 18 ⊢ (n = p → ((( Spac ‘n) ∈ Fin → Tc p ≤c n) ↔ (( Spac ‘p) ∈ Fin → Tc p ≤c p)))
65	64	rspcv 2951	. . . . . . . . . . . . . . . . 17 ⊢ (p ∈ NC → (∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n) → (( Spac ‘p) ∈ Fin → Tc p ≤c p)))
66	65	imp 418	. . . . . . . . . . . . . . . 16 ⊢ ((p ∈ NC ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n)) → (( Spac ‘p) ∈ Fin → Tc p ≤c p))
67	66	ad2ant2l 726	. . . . . . . . . . . . . . 15 ⊢ ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n))) → (( Spac ‘p) ∈ Fin → Tc p ≤c p))
68	60, 67	mpd 14	. . . . . . . . . . . . . 14 ⊢ ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n))) → Tc p ≤c p)
69		simplr 731	. . . . . . . . . . . . . . . 16 ⊢ ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n))) → p ∈ NC )
70		tccl 6160	. . . . . . . . . . . . . . . 16 ⊢ (p ∈ NC → Tc p ∈ NC )
71	69, 70	syl 15	. . . . . . . . . . . . . . 15 ⊢ ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n))) → Tc p ∈ NC )
72		tlecg 6230	. . . . . . . . . . . . . . 15 ⊢ (( Tc p ∈ NC ∧ p ∈ NC ) → ( Tc p ≤c p ↔ Tc Tc p ≤c Tc p))
73	71, 69, 72	syl2anc 642	. . . . . . . . . . . . . 14 ⊢ ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n))) → ( Tc p ≤c p ↔ Tc Tc p ≤c Tc p))
74	68, 73	mpbid 201	. . . . . . . . . . . . 13 ⊢ ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n))) → Tc Tc p ≤c Tc p)
75		fveq2 5328	. . . . . . . . . . . . . . . . 17 ⊢ (m = Tc p → ( Spac ‘m) = ( Spac ‘ Tc p))
76	75	eleq1d 2419	. . . . . . . . . . . . . . . 16 ⊢ (m = Tc p → (( Spac ‘m) ∈ Fin ↔ ( Spac ‘ Tc p) ∈ Fin ))
77		breq1 4642	. . . . . . . . . . . . . . . . . 18 ⊢ (m = Tc p → (m ≤c n ↔ Tc p ≤c n))
78	77	imbi2d 307	. . . . . . . . . . . . . . . . 17 ⊢ (m = Tc p → ((( Spac ‘n) ∈ Fin → m ≤c n) ↔ (( Spac ‘n) ∈ Fin → Tc p ≤c n)))
79	78	ralbidv 2634	. . . . . . . . . . . . . . . 16 ⊢ (m = Tc p → (∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n) ↔ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n)))
80	76, 79	anbi12d 691	. . . . . . . . . . . . . . 15 ⊢ (m = Tc p → ((( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n)) ↔ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n))))
81	80	anbi2d 684	. . . . . . . . . . . . . 14 ⊢ (m = Tc p → ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) ↔ (( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n)))))
82		tceq 6158	. . . . . . . . . . . . . . 15 ⊢ (m = Tc p → Tc m = Tc Tc p)
83		id 19	. . . . . . . . . . . . . . 15 ⊢ (m = Tc p → m = Tc p)
84	82, 83	breq12d 4652	. . . . . . . . . . . . . 14 ⊢ (m = Tc p → ( Tc m ≤c m ↔ Tc Tc p ≤c Tc p))
85	81, 84	imbi12d 311	. . . . . . . . . . . . 13 ⊢ (m = Tc p → (((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → Tc m ≤c m) ↔ ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘ Tc p) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → Tc p ≤c n))) → Tc Tc p ≤c Tc p)))
86	74, 85	mpbiri 224	. . . . . . . . . . . 12 ⊢ (m = Tc p → ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → Tc m ≤c m))
87	86	com12 27	. . . . . . . . . . 11 ⊢ ((( ≤c We NC ∧ p ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → (m = Tc p → Tc m ≤c m))
88	87	an32s 779	. . . . . . . . . 10 ⊢ ((( ≤c We NC ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) ∧ p ∈ NC ) → (m = Tc p → Tc m ≤c m))
89	88	rexlimdva 2738	. . . . . . . . 9 ⊢ (( ≤c We NC ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → (∃p ∈ NC m = Tc p → Tc m ≤c m))
90	89	adantlr 695	. . . . . . . 8 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → (∃p ∈ NC m = Tc p → Tc m ≤c m))
91	58, 90	syld 40	. . . . . . 7 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → (m ≤c Tc m → Tc m ≤c m))
92	55, 91	mpd 14	. . . . . 6 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → Tc m ≤c m)
93	41, 44, 42, 92, 55	antid 5929	. . . . 5 ⊢ ((( ≤c We NC ∧ m ∈ NC ) ∧ (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n))) → Tc m = m)
94	93	exp32 588	. . . 4 ⊢ (( ≤c We NC ∧ m ∈ NC ) → (( Spac ‘m) ∈ Fin → (∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n) → Tc m = m)))
95	94	imdistand 673	. . 3 ⊢ (( ≤c We NC ∧ m ∈ NC ) → ((( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n)) → (( Spac ‘m) ∈ Fin ∧ Tc m = m)))
96	95	reximdva 2726	. 2 ⊢ ( ≤c We NC → (∃m ∈ NC (( Spac ‘m) ∈ Fin ∧ ∀n ∈ NC (( Spac ‘n) ∈ Fin → m ≤c n)) → ∃m ∈ NC (( Spac ‘m) ∈ Fin ∧ Tc m = m)))
97	28, 96	mpd 14	1 ⊢ ( ≤c We NC → ∃m ∈ NC (( Spac ‘m) ∈ Fin ∧ Tc m = m))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ∩ cin 3208  ℘cpw 3722  {csn 3737  1_cc1c 4134  ℘₁cpw1 4135  0_cc0c 4374   +_c cplc 4375   Fin cfin 4376   class class class wbr 4639   ‘cfv 4781  (class class class)co 5525   Trans ctrans 5888   Ref cref 5889   Antisym cantisym 5890   Po cpartial 5891   Connex cconnex 5892   Or cstrict 5893   Fr cfound 5894   We cwe 5895   NC cncs 6088   ≤_c clec 6089   <_c cltc 6090   Nc cnc 6091   T_c ctc 6093  2_cc2c 6094   ↑_c cce 6096   Sp_ac cspac 6273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-meredith 1406  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-tp 3743  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-fix 5740  df-cup 5742  df-disj 5744  df-addcfn 5746  df-compose 5748  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-fullfun 5768  df-clos1 5873  df-trans 5899  df-ref 5900  df-antisym 5901  df-partial 5902  df-connex 5903  df-strict 5904  df-found 5905  df-we 5906  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-lec 6099  df-ltc 6100  df-nc 6101  df-tc 6103  df-2c 6104  df-3c 6105  df-ce 6106  df-tcfn 6107  df-spac 6274

This theorem is referenced by:  nchoice  6308