Theorem rankxplim3 5825

Description: The rank of a cross product is a limit ordinal iff its union is.

Hypotheses

Ref	Expression
rankxplim.1	|- A e. _V
rankxplim.2	|- B e. _V

Assertion

Ref	Expression
rankxplim3	|- (Lim (rank` (A X. B)) <-> Lim U.(rank` (A X. B)))

Proof of Theorem rankxplim3

Step	Hyp	Ref	Expression
1		limuni2 3725	. 2 |- (Lim (rank` (A X. B)) -> Lim U.(rank` (A X. B)))
2		0ellim 3726	. . . 4 |- (Lim U.(rank` (A X. B)) -> (/) e. U.(rank` (A X. B)))
3		n0i 2880	. . . 4 |- ((/) e. U.(rank` (A X. B)) -> -. U.(rank` (A X. B)) = (/))
4		unieq 3185	. . . . . 6 |- ((rank` (A X. B)) = (/) -> U.(rank` (A X. B)) = U.(/))
5		uni0 3205	. . . . . 6 |- U.(/) = (/)
6	4, 5	syl6eq 1944	. . . . 5 |- ((rank` (A X. B)) = (/) -> U.(rank` (A X. B)) = (/))
7	6	con3i 114	. . . 4 |- (-. U.(rank` (A X. B)) = (/) -> -. (rank` (A X. B)) = (/))
8	2, 3, 7	3syl 24	. . 3 |- (Lim U.(rank` (A X. B)) -> -. (rank` (A X. B)) = (/))
9		rankon 5782	. . . . . . . . . 10 |- (rank` (A u. B)) e. On
10	9	onsuci 3919	. . . . . . . . 9 |- suc (rank` (A u. B)) e. On
11	10	onsuci 3919	. . . . . . . 8 |- suc suc (rank` (A u. B)) e. On
12	11	elisseti 2301	. . . . . . 7 |- suc suc (rank` (A u. B)) e. _V
13	12	sucid 3744	. . . . . 6 |- suc suc (rank` (A u. B)) e. suc suc suc (rank` (A u. B))
14	11	onsuci 3919	. . . . . . . 8 |- suc suc suc (rank` (A u. B)) e. On
15		ontri1 3695	. . . . . . . 8 |- ((suc suc suc (rank` (A u. B)) e. On /\ suc suc (rank` (A u. B)) e. On) -> (suc suc suc (rank` (A u. B)) C_ suc suc (rank` (A u. B)) <-> -. suc suc (rank` (A u. B)) e. suc suc suc (rank` (A u. B))))
16	14, 11, 15	mp2an 761	. . . . . . 7 |- (suc suc suc (rank` (A u. B)) C_ suc suc (rank` (A u. B)) <-> -. suc suc (rank` (A u. B)) e. suc suc suc (rank` (A u. B)))
17	16	con2bii 238	. . . . . 6 |- (suc suc (rank` (A u. B)) e. suc suc suc (rank` (A u. B)) <-> -. suc suc suc (rank` (A u. B)) C_ suc suc (rank` (A u. B)))
18	13, 17	mpbi 206	. . . . 5 |- -. suc suc suc (rank` (A u. B)) C_ suc suc (rank` (A u. B))
19		rankxplim.1	. . . . . . 7 |- A e. _V
20		rankxplim.2	. . . . . . 7 |- B e. _V
21	19, 20	rankxpu 5822	. . . . . 6 |- (rank` (A X. B)) C_ suc suc (rank` (A u. B))
22		sstr 2625	. . . . . 6 |- ((suc suc suc (rank` (A u. B)) C_ (rank` (A X. B)) /\ (rank` (A X. B)) C_ suc suc (rank` (A u. B))) -> suc suc suc (rank` (A u. B)) C_ suc suc (rank` (A u. B)))
23	21, 22	mpan2 760	. . . . 5 |- (suc suc suc (rank` (A u. B)) C_ (rank` (A X. B)) -> suc suc suc (rank` (A u. B)) C_ suc suc (rank` (A u. B)))
24	18, 23	mto 121	. . . 4 |- -. suc suc suc (rank` (A u. B)) C_ (rank` (A X. B))
25		simprl 450	. . . . . . . . . . . . 13 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> (rank` (A u. B)) = suc x)
26		simpr 350	. . . . . . . . . . . . . . . . . 18 |- ((Lim U.(rank` (A X. B)) /\ (rank` (A u. B)) = suc x) -> (rank` (A u. B)) = suc x)
27		df-ne 2019	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((A X. B) =/= (/) <-> -. (A X. B) = (/))
28	19, 20	xpex 4096	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (A X. B) e. _V
29	28	rankeq0 5807	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((A X. B) = (/) <-> (rank` (A X. B)) = (/))
30	29	notbii 204	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (-. (A X. B) = (/) <-> -. (rank` (A X. B)) = (/))
31	27, 30	bitr2i 191	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (-. (rank` (A X. B)) = (/) <-> (A X. B) =/= (/))
32	8, 31	sylib 215	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Lim U.(rank` (A X. B)) -> (A X. B) =/= (/))
33		unixp 4422	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((A X. B) =/= (/) -> U.U.(A X. B) = (A u. B))
34	32, 33	syl 12	. . . . . . . . . . . . . . . . . . . . . 22 |- (Lim U.(rank` (A X. B)) -> U.U.(A X. B) = (A u. B))
35	34	fveq2d 4685	. . . . . . . . . . . . . . . . . . . . 21 |- (Lim U.(rank` (A X. B)) -> (rank` U.U.(A X. B)) = (rank` (A u. B)))
36		rankuni 5809	. . . . . . . . . . . . . . . . . . . . . 22 |- (rank` U.U.(A X. B)) = U.(rank` U.(A X. B))
37		rankuni 5809	. . . . . . . . . . . . . . . . . . . . . . 23 |- (rank` U.(A X. B)) = U.(rank` (A X. B))
38	37	unieqi 3187	. . . . . . . . . . . . . . . . . . . . . 22 |- U.(rank` U.(A X. B)) = U.U.(rank` (A X. B))
39	36, 38	eqtri 1908	. . . . . . . . . . . . . . . . . . . . 21 |- (rank` U.U.(A X. B)) = U.U.(rank` (A X. B))
40	35, 39	syl5reqr 1943	. . . . . . . . . . . . . . . . . . . 20 |- (Lim U.(rank` (A X. B)) -> (rank` (A u. B)) = U.U.(rank` (A X. B)))
41		eqimss 2665	. . . . . . . . . . . . . . . . . . . 20 |- ((rank` (A u. B)) = U.U.(rank` (A X. B)) -> (rank` (A u. B)) C_ U.U.(rank` (A X. B)))
42	40, 41	syl 12	. . . . . . . . . . . . . . . . . . 19 |- (Lim U.(rank` (A X. B)) -> (rank` (A u. B)) C_ U.U.(rank` (A X. B)))
43	42	adantr 425	. . . . . . . . . . . . . . . . . 18 |- ((Lim U.(rank` (A X. B)) /\ (rank` (A u. B)) = suc x) -> (rank` (A u. B)) C_ U.U.(rank` (A X. B)))
44	26, 43	eqsstr3d 2652	. . . . . . . . . . . . . . . . 17 |- ((Lim U.(rank` (A X. B)) /\ (rank` (A u. B)) = suc x) -> suc x C_ U.U.(rank` (A X. B)))
45	44	adantrr 431	. . . . . . . . . . . . . . . 16 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> suc x C_ U.U.(rank` (A X. B)))
46		limuni 3724	. . . . . . . . . . . . . . . . 17 |- (Lim U.(rank` (A X. B)) -> U.(rank` (A X. B)) = U.U.(rank` (A X. B)))
47	46	adantr 425	. . . . . . . . . . . . . . . 16 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> U.(rank` (A X. B)) = U.U.(rank` (A X. B)))
48	45, 47	sseqtr4d 2654	. . . . . . . . . . . . . . 15 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> suc x C_ U.(rank` (A X. B)))
49		visset 2295	. . . . . . . . . . . . . . . 16 |- x e. _V
50		rankon 5782	. . . . . . . . . . . . . . . . . 18 |- (rank` (A X. B)) e. On
51	50	onordi 3774	. . . . . . . . . . . . . . . . 17 |- Ord (rank` (A X. B))
52		orduni 3875	. . . . . . . . . . . . . . . . 17 |- (Ord (rank` (A X. B)) -> Ord U.(rank` (A X. B)))
53	51, 52	ax-mp 7	. . . . . . . . . . . . . . . 16 |- Ord U.(rank` (A X. B))
54		ordelsuc 3900	. . . . . . . . . . . . . . . 16 |- ((x e. _V /\ Ord U.(rank` (A X. B))) -> (x e. U.(rank` (A X. B)) <-> suc x C_ U.(rank` (A X. B))))
55	49, 53, 54	mp2an 761	. . . . . . . . . . . . . . 15 |- (x e. U.(rank` (A X. B)) <-> suc x C_ U.(rank` (A X. B)))
56	48, 55	sylibr 217	. . . . . . . . . . . . . 14 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> x e. U.(rank` (A X. B)))
57		limsuc 3933	. . . . . . . . . . . . . . 15 |- (Lim U.(rank` (A X. B)) -> (x e. U.(rank` (A X. B)) <-> suc x e. U.(rank` (A X. B))))
58	57	adantr 425	. . . . . . . . . . . . . 14 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> (x e. U.(rank` (A X. B)) <-> suc x e. U.(rank` (A X. B))))
59	56, 58	mpbid 212	. . . . . . . . . . . . 13 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> suc x e. U.(rank` (A X. B)))
60	25, 59	eqeltrd 1971	. . . . . . . . . . . 12 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> (rank` (A u. B)) e. U.(rank` (A X. B)))
61		limsuc 3933	. . . . . . . . . . . . 13 |- (Lim U.(rank` (A X. B)) -> ((rank` (A u. B)) e. U.(rank` (A X. B)) <-> suc (rank` (A u. B)) e. U.(rank` (A X. B))))
62	61	adantr 425	. . . . . . . . . . . 12 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> ((rank` (A u. B)) e. U.(rank` (A X. B)) <-> suc (rank` (A u. B)) e. U.(rank` (A X. B))))
63	60, 62	mpbid 212	. . . . . . . . . . 11 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> suc (rank` (A u. B)) e. U.(rank` (A X. B)))
64		ordsucelsuc 3902	. . . . . . . . . . . 12 |- (Ord U.(rank` (A X. B)) -> (suc (rank` (A u. B)) e. U.(rank` (A X. B)) <-> suc suc (rank` (A u. B)) e. suc U.(rank` (A X. B))))
65	53, 64	ax-mp 7	. . . . . . . . . . 11 |- (suc (rank` (A u. B)) e. U.(rank` (A X. B)) <-> suc suc (rank` (A u. B)) e. suc U.(rank` (A X. B)))
66	63, 65	sylib 215	. . . . . . . . . 10 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> suc suc (rank` (A u. B)) e. suc U.(rank` (A X. B)))
67		onsucuni2 3914	. . . . . . . . . . . 12 |- (((rank` (A X. B)) e. On /\ (rank` (A X. B)) = suc y) -> suc U.(rank` (A X. B)) = (rank` (A X. B)))
68	50, 67	mpan 759	. . . . . . . . . . 11 |- ((rank` (A X. B)) = suc y -> suc U.(rank` (A X. B)) = (rank` (A X. B)))
69	68	ad2antll 443	. . . . . . . . . 10 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> suc U.(rank` (A X. B)) = (rank` (A X. B)))
70	66, 69	eleqtrd 1973	. . . . . . . . 9 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> suc suc (rank` (A u. B)) e. (rank` (A X. B)))
71	11, 50	onsucssi 3922	. . . . . . . . 9 |- (suc suc (rank` (A u. B)) e. (rank` (A X. B)) <-> suc suc suc (rank` (A u. B)) C_ (rank` (A X. B)))
72	70, 71	sylib 215	. . . . . . . 8 |- ((Lim U.(rank` (A X. B)) /\ ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y)) -> suc suc suc (rank` (A u. B)) C_ (rank` (A X. B)))
73	72	ex 402	. . . . . . 7 |- (Lim U.(rank` (A X. B)) -> (((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y) -> suc suc suc (rank` (A u. B)) C_ (rank` (A X. B))))
74	73	a1d 15	. . . . . 6 |- (Lim U.(rank` (A X. B)) -> ((x e. On /\ y e. On) -> (((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y) -> suc suc suc (rank` (A u. B)) C_ (rank` (A X. B)))))
75	74	r19.23advv 2218	. . . . 5 |- (Lim U.(rank` (A X. B)) -> (E.x e. On E.y e. On ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y) -> suc suc suc (rank` (A u. B)) C_ (rank` (A X. B))))
76		reeanv 2249	. . . . 5 |- (E.x e. On E.y e. On ((rank` (A u. B)) = suc x /\ (rank` (A X. B)) = suc y) <-> (E.x e. On (rank` (A u. B)) = suc x /\ E.y e. On (rank` (A X. B)) = suc y))
77	75, 76	syl5ibr 224	. . . 4 |- (Lim U.(rank` (A X. B)) -> ((E.x e. On (rank` (A u. B)) = suc x /\ E.y e. On (rank` (A X. B)) = suc y) -> suc suc suc (rank` (A u. B)) C_ (rank` (A X. B))))
78	24, 77	mtoi 122	. . 3 |- (Lim U.(rank` (A X. B)) -> -. (E.x e. On (rank` (A u. B)) = suc x /\ E.y e. On (rank` (A X. B)) = suc y))
79	9	onordi 3774	. . . . . . . . . . 11 |- Ord (rank` (A u. B))
80		ordzsl 3927	. . . . . . . . . . 11 |- (Ord (rank` (A u. B)) <-> ((rank` (A u. B)) = (/) \/ E.x e. On (rank` (A u. B)) = suc x \/ Lim (rank` (A u. B))))
81	79, 80	mpbi 206	. . . . . . . . . 10 |- ((rank` (A u. B)) = (/) \/ E.x e. On (rank` (A u. B)) = suc x \/ Lim (rank` (A u. B)))
82	81	3ori 1157	. . . . . . . . 9 |- ((-. (rank` (A u. B)) = (/) /\ -. E.x e. On (rank` (A u. B)) = suc x) -> Lim (rank` (A u. B)))
83		un00 2907	. . . . . . . . . . . . 13 |- ((A = (/) /\ B = (/)) <-> (A u. B) = (/))
84		olc 290	. . . . . . . . . . . . . 14 |- (B = (/) -> (A = (/) \/ B = (/)))
85	84	adantl 424	. . . . . . . . . . . . 13 |- ((A = (/) /\ B = (/)) -> (A = (/) \/ B = (/)))
86	83, 85	sylbir 218	. . . . . . . . . . . 12 |- ((A u. B) = (/) -> (A = (/) \/ B = (/)))
87		xpeq0 4336	. . . . . . . . . . . 12 |- ((A X. B) = (/) <-> (A = (/) \/ B = (/)))
88	86, 87	sylibr 217	. . . . . . . . . . 11 |- ((A u. B) = (/) -> (A X. B) = (/))
89	88	con3i 114	. . . . . . . . . 10 |- (-. (A X. B) = (/) -> -. (A u. B) = (/))
90	19, 20	unex 3796	. . . . . . . . . . . 12 |- (A u. B) e. _V
91	90	rankeq0 5807	. . . . . . . . . . 11 |- ((A u. B) = (/) <-> (rank` (A u. B)) = (/))
92	91	notbii 204	. . . . . . . . . 10 |- (-. (A u. B) = (/) <-> -. (rank` (A u. B)) = (/))
93	89, 30, 92	3imtr3i 235	. . . . . . . . 9 |- (-. (rank` (A X. B)) = (/) -> -. (rank` (A u. B)) = (/))
94	82, 93	sylan 497	. . . . . . . 8 |- ((-. (rank` (A X. B)) = (/) /\ -. E.x e. On (rank` (A u. B)) = suc x) -> Lim (rank` (A u. B)))
95	94	ex 402	. . . . . . 7 |- (-. (rank` (A X. B)) = (/) -> (-. E.x e. On (rank` (A u. B)) = suc x -> Lim (rank` (A u. B))))
96		ordzsl 3927	. . . . . . . . . 10 |- (Ord (rank` (A X. B)) <-> ((rank` (A X. B)) = (/) \/ E.y e. On (rank` (A X. B)) = suc y \/ Lim (rank` (A X. B))))
97	51, 96	mpbi 206	. . . . . . . . 9 |- ((rank` (A X. B)) = (/) \/ E.y e. On (rank` (A X. B)) = suc y \/ Lim (rank` (A X. B)))
98	97	3ori 1157	. . . . . . . 8 |- ((-. (rank` (A X. B)) = (/) /\ -. E.y e. On (rank` (A X. B)) = suc y) -> Lim (rank` (A X. B)))
99	98	ex 402	. . . . . . 7 |- (-. (rank` (A X. B)) = (/) -> (-. E.y e. On (rank` (A X. B)) = suc y -> Lim (rank` (A X. B))))
100	95, 99	orim12d 624	. . . . . 6 |- (-. (rank` (A X. B)) = (/) -> ((-. E.x e. On (rank` (A u. B)) = suc x \/ -. E.y e. On (rank` (A X. B)) = suc y) -> (Lim (rank` (A u. B)) \/ Lim (rank` (A X. B)))))
101		ianor 329	. . . . . 6 |- (-. (E.x e. On (rank` (A u. B)) = suc x /\ E.y e. On (rank` (A X. B)) = suc y) <-> (-. E.x e. On (rank` (A u. B)) = suc x \/ -. E.y e. On (rank` (A X. B)) = suc y))
102	100, 101	syl5ib 223	. . . . 5 |- (-. (rank` (A X. B)) = (/) -> (-. (E.x e. On (rank` (A u. B)) = suc x /\ E.y e. On (rank` (A X. B)) = suc y) -> (Lim (rank` (A u. B)) \/ Lim (rank` (A X. B)))))
103	102	imp 377	. . . 4 |- ((-. (rank` (A X. B)) = (/) /\ -. (E.x e. On (rank` (A u. B)) = suc x /\ E.y e. On (rank` (A X. B)) = suc y)) -> (Lim (rank` (A u. B)) \/ Lim (rank` (A X. B))))
104		simpl 346	. . . . . . . 8 |- ((Lim (rank` (A u. B)) /\ -. (rank` (A X. B)) = (/)) -> Lim (rank` (A u. B)))
105	19, 20	rankxplim 5823	. . . . . . . . . 10 |- ((Lim (rank` (A u. B)) /\ (A X. B) =/= (/)) -> (rank` (A X. B)) = (rank` (A u. B)))
106	29	necon3abii 2030	. . . . . . . . . 10 |- ((A X. B) =/= (/) <-> -. (rank` (A X. B)) = (/))
107	105, 106	sylan2br 502	. . . . . . . . 9 |- ((Lim (rank` (A u. B)) /\ -. (rank` (A X. B)) = (/)) -> (rank` (A X. B)) = (rank` (A u. B)))
108		limeq 3669	. . . . . . . . 9 |- ((rank` (A X. B)) = (rank` (A u. B)) -> (Lim (rank` (A X. B)) <-> Lim (rank` (A u. B))))
109	107, 108	syl 12	. . . . . . . 8 |- ((Lim (rank` (A u. B)) /\ -. (rank` (A X. B)) = (/)) -> (Lim (rank` (A X. B)) <-> Lim (rank` (A u. B))))
110	104, 109	mpbird 213	. . . . . . 7 |- ((Lim (rank` (A u. B)) /\ -. (rank` (A X. B)) = (/)) -> Lim (rank` (A X. B)))
111	110	expcom 403	. . . . . 6 |- (-. (rank` (A X. B)) = (/) -> (Lim (rank` (A u. B)) -> Lim (rank` (A X. B))))
112		idd 75	. . . . . 6 |- (-. (rank` (A X. B)) = (/) -> (Lim (rank` (A X. B)) -> Lim (rank` (A X. B))))
113	111, 112	jaod 469	. . . . 5 |- (-. (rank` (A X. B)) = (/) -> ((Lim (rank` (A u. B)) \/ Lim (rank` (A X. B))) -> Lim (rank` (A X. B))))
114	113	adantr 425	. . . 4 |- ((-. (rank` (A X. B)) = (/) /\ -. (E.x e. On (rank` (A u. B)) = suc x /\ E.y e. On (rank` (A X. B)) = suc y)) -> ((Lim (rank` (A u. B)) \/ Lim (rank` (A X. B))) -> Lim (rank` (A X. B))))
115	103, 114	mpd 29	. . 3 |- ((-. (rank` (A X. B)) = (/) /\ -. (E.x e. On (rank` (A u. B)) = suc x /\ E.y e. On (rank` (A X. B)) = suc y)) -> Lim (rank` (A X. B)))
116	8, 78, 115	syl11anc 524	. 2 |- (Lim U.(rank` (A X. B)) -> Lim (rank` (A X. B)))
117	1, 116	impbii 174	1 |- (Lim (rank` (A X. B)) <-> Lim U.(rank` (A X. B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106  _Vcvv 2292   u. cun 2591   C_ wss 2593  (/)c0 2875  U.cuni 3177  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659   X. cxp 3984  ` cfv 3998  rankcrnk 5749

This theorem is referenced by:  rankxpsuc 5826

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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