Theorem nchoicelem19 6308

Description: Lemma for nchoice 6309. 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 6307	. . 3 ⊢ {x ∣ ( Spac ‘x) ∈ Fin } ∈ V
2		fveq2 5329	. . . 4 ⊢ (x = m → ( Spac ‘x) = ( Spac ‘m))
3	2	eleq1d 2419	. . 3 ⊢ (x = m → (( Spac ‘x) ∈ Fin ↔ ( Spac ‘m) ∈ Fin ))
4		fveq2 5329	. . . 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 4110	. . . . 5 ⊢ V ∈ V
8	7	ncelncsi 6122	. . . 4 ⊢ Nc V ∈ NC
9		ltcpw1pwg 6203	. . . . . . . . 9 ⊢ (V ∈ V → Nc ℘1V <c Nc ℘V)
10	7, 9	ax-mp 5	. . . . . . . 8 ⊢ Nc ℘1V <c Nc ℘V
11		df1c2 4169	. . . . . . . . 9 ⊢ 1c = ℘1V
12	11	nceqi 6110	. . . . . . . 8 ⊢ Nc 1c = Nc ℘1V
13		pwv 3887	. . . . . . . . . 10 ⊢ ℘V = V
14	13	nceqi 6110	. . . . . . . . 9 ⊢ Nc ℘V = Nc V
15	14	eqcomi 2357	. . . . . . . 8 ⊢ Nc V = Nc ℘V
16	10, 12, 15	3brtr4i 4668	. . . . . . 7 ⊢ Nc 1c <c Nc V
17		nchoicelem8 6297	. . . . . . . 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 6292	. . . . . 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 4432	. . . . 5 ⊢ { Nc V} ∈ Fin
23	21, 22	syl6eqel 2441	. . . 4 ⊢ ( ≤c We NC → ( Spac ‘ Nc V) ∈ Fin )
24		fveq2 5329	. . . . . 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 2956	. . . 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 5939	. 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 5907	. . . . . . . . . . 11 ⊢ We = ( Or ∩ Fr )
31	30	breqi 4646	. . . . . . . . . 10 ⊢ ( ≤c We NC ↔ ≤c ( Or ∩ Fr ) NC )
32		brin 4694	. . . . . . . . . 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 5935	. . . . . . . . . 10 ⊢ ( ≤c Or NC ↔ ( ≤c Po NC ∧ ≤c Connex NC ))
36	35	simplbi 446	. . . . . . . . 9 ⊢ ( ≤c Or NC → ≤c Po NC )
37		porta 5934	. . . . . . . . . 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 6161	. . . . . . 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 6306	. . . . . . . . . 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 5329	. . . . . . . . . . 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 4644	. . . . . . . . . 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 2952	. . . . . . . 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 6232	. . . . . . . . . 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 6301	. . . . . . . . . . . . . . . 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 5329	. . . . . . . . . . . . . . . . . . . 20 ⊢ (n = p → ( Spac ‘n) = ( Spac ‘p))
62	61	eleq1d 2419	. . . . . . . . . . . . . . . . . . 19 ⊢ (n = p → (( Spac ‘n) ∈ Fin ↔ ( Spac ‘p) ∈ Fin ))
63		breq2 4644	. . . . . . . . . . . . . . . . . . 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 2952	. . . . . . . . . . . . . . . . 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 6161	. . . . . . . . . . . . . . . 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 6231	. . . . . . . . . . . . . . 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 5329	. . . . . . . . . . . . . . . . 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 4643	. . . . . . . . . . . . . . . . . 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 2635	. . . . . . . . . . . . . . . 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 6159	. . . . . . . . . . . . . . 15 ⊢ (m = Tc p → Tc m = Tc Tc p)
83		id 19	. . . . . . . . . . . . . . 15 ⊢ (m = Tc p → m = Tc p)
84	82, 83	breq12d 4653	. . . . . . . . . . . . . 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 2739	. . . . . . . . 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 5930	. . . . 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 2727	. 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 2615  ∃wrex 2616  Vcvv 2860   ∩ cin 3209  ℘cpw 3723  {csn 3738  1_cc1c 4135  ℘₁cpw1 4136  0_cc0c 4375   +_c cplc 4376   Fin cfin 4377   class class class wbr 4640   ‘cfv 4782  (class class class)co 5526   Trans ctrans 5889   Ref cref 5890   Antisym cantisym 5891   Po cpartial 5892   Connex cconnex 5893   Or cstrict 5894   Fr cfound 5895   We cwe 5896   NC cncs 6089   ≤_c clec 6090   <_c cltc 6091   Nc cnc 6092   T_c ctc 6094  2_cc2c 6095   ↑_c cce 6097   Sp_ac cspac 6274

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-tp 3744  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-cup 5743  df-disj 5745  df-addcfn 5747  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-fullfun 5769  df-clos1 5874  df-trans 5900  df-ref 5901  df-antisym 5902  df-partial 5903  df-connex 5904  df-strict 5905  df-found 5906  df-we 5907  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-lec 6100  df-ltc 6101  df-nc 6102  df-tc 6104  df-2c 6105  df-3c 6106  df-ce 6107  df-tcfn 6108  df-spac 6275

This theorem is referenced by:  nchoice  6309