Theorem infxpidmlem12 7775

Description: Lemma for infxpidm 7776. Letting x be the range of a maximal bijection g in H, infxpidmlem11 7774 lets us show that assuming x ~<_ (A \ x) leads to the contradiction E.h e. Hg (. h meaning g is not maximal. Thus (A \ x) ~< x. This allows us to show that x is as big as A. Since x is idempotent, and g exists by Zorn's Lemma in infxpidmlem9 7772, A is also idempotent.

Hypotheses

Ref	Expression
infxpidmlem.1	|- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
infxpidmlem.2	|- A e. V

Assertion

Ref	Expression
infxpidmlem12	|- (om ~<_ A -> (A X. A) ~~ A)

Distinct variable group:   t,f,A

Proof of Theorem infxpidmlem12

Step	Hyp	Ref	Expression
1		infxpidmlem.1	. . 3 |- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t (_ A) /\ f:(t X. t)-1-1-onto->t))}
2		infxpidmlem.2	. . 3 |- A e. V
3	1, 2	infxpidmlem9 7772	. 2 |- E.g e. H A.h e. H -. g (. h
4	1, 2	infxpidmlem10 7773	. . . . 5 |- (A.h e. H -. g (. h -> (om ~<_ A -> g =/= (/)))
5		visset 1859	. . . . . . . . 9 |- g e. V
6	1, 5	infxpidmlem2 7765	. . . . . . . 8 |- (g e. H <-> (g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
7		neor 1684	. . . . . . . 8 |- ((g = (/) \/ E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)) <-> (g =/= (/) -> E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
8	6, 7	bitri 171	. . . . . . 7 |- (g e. H <-> (g =/= (/) -> E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)))
9		entr 4555	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (x X. y)) -> x ~~ (x X. y))
10		visset 1859	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- x e. V
11	10	enref 4532	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- x ~~ x
12		visset 1859	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- y e. V
13	10, 10, 10, 12	xpen 4635	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((x ~~ x /\ x ~~ y) -> (x X. x) ~~ (x X. y))
14	11, 13	mpan 699	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x ~~ y -> (x X. x) ~~ (x X. y))
15	9, 14	sylan2 453	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((x ~~ (x X. x) /\ x ~~ y) -> x ~~ (x X. y))
16	15	adantll 392	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> x ~~ (x X. y))
17		entr 4555	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (y X. x)) -> x ~~ (y X. x))
18	10, 12, 10, 10	xpen 4635	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((x ~~ y /\ x ~~ x) -> (x X. x) ~~ (y X. x))
19	11, 18	mpan2 700	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x ~~ y -> (x X. x) ~~ (y X. x))
20	17, 19	sylan2 453	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((x ~~ (x X. x) /\ x ~~ y) -> x ~~ (y X. x))
21		entr 4555	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (y X. y)) -> x ~~ (y X. y))
22	10, 12, 10, 12	xpen 4635	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((x ~~ y /\ x ~~ y) -> (x X. x) ~~ (y X. y))
23	22	anidms 435	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x ~~ y -> (x X. x) ~~ (y X. y))
24	21, 23	sylan2 453	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((x ~~ (x X. x) /\ x ~~ y) -> x ~~ (y X. y))
25	20, 24	jca 286	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((x ~~ (x X. x) /\ x ~~ y) -> (x ~~ (y X. x) /\ x ~~ (y X. y)))
26	25	adantll 392	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> (x ~~ (y X. x) /\ x ~~ (y X. y)))
27	12, 10	xpex 3349	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y X. x) e. V
28	12, 12	xpex 3349	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y X. y) e. V
29	27, 28	infxpidmlem1 7764	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((om ~<_ x /\ x ~~ (x X. x)) -> ((x ~~ (y X. x) /\ x ~~ (y X. y)) -> x ~~ ((y X. x) u. (y X. y))))
30	29	adantr 389	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> ((x ~~ (y X. x) /\ x ~~ (y X. y)) -> x ~~ ((y X. x) u. (y X. y))))
31	26, 30	mpd 26	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> x ~~ ((y X. x) u. (y X. y)))
32	10, 12	xpex 3349	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x X. y) e. V
33	27, 28	unex 3095	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((y X. x) u. (y X. y)) e. V
34	32, 33	infxpidmlem1 7764	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((om ~<_ x /\ x ~~ (x X. x)) -> ((x ~~ (x X. y) /\ x ~~ ((y X. x) u. (y X. y))) -> x ~~ ((x X. y) u. ((y X. x) u. (y X. y)))))
35	34	adantr 389	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> ((x ~~ (x X. y) /\ x ~~ ((y X. x) u. (y X. y))) -> x ~~ ((x X. y) u. ((y X. x) u. (y X. y)))))
36	16, 31, 35	mp2and 707	. . . . . . . . . . . . . . . . . . . . . 22 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> x ~~ ((x X. y) u. ((y X. x) u. (y X. y))))
37	32, 33	unex 3095	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((x X. y) u. ((y X. x) u. (y X. y))) e. V
38	37	ensym 4553	. . . . . . . . . . . . . . . . . . . . . 22 |- (x ~~ ((x X. y) u. ((y X. x) u. (y X. y))) -> ((x X. y) u. ((y X. x) u. (y X. y))) ~~ x)
39	36, 38	syl 10	. . . . . . . . . . . . . . . . . . . . 21 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> ((x X. y) u. ((y X. x) u. (y X. y))) ~~ x)
40		entr 4555	. . . . . . . . . . . . . . . . . . . . 21 |- ((((x X. y) u. ((y X. x) u. (y X. y))) ~~ x /\ x ~~ y) -> ((x X. y) u. ((y X. x) u. (y X. y))) ~~ y)
41	39, 40	sylancom 478	. . . . . . . . . . . . . . . . . . . 20 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> ((x X. y) u. ((y X. x) u. (y X. y))) ~~ y)
42	10	ensym 4553	. . . . . . . . . . . . . . . . . . . 20 |- ((x X. x) ~~ x -> x ~~ (x X. x))
43	41, 42	sylanl2 463	. . . . . . . . . . . . . . . . . . 19 |- (((om ~<_ x /\ (x X. x) ~~ x) /\ x ~~ y) -> ((x X. y) u. ((y X. x) u. (y X. y))) ~~ y)
44	12	bren 4518	. . . . . . . . . . . . . . . . . . 19 |- (((x X. y) u. ((y X. x) u. (y X. y))) ~~ y <-> E.u u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y)
45	43, 44	sylib 196	. . . . . . . . . . . . . . . . . 18 |- (((om ~<_ x /\ (x X. x) ~~ x) /\ x ~~ y) -> E.u u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y)
46	10, 10	xpex 3349	. . . . . . . . . . . . . . . . . . . . 21 |- (x X. x) e. V
47	46	f1oen 4539	. . . . . . . . . . . . . . . . . . . 20 |- (g:(x X. x)-1-1-onto->x -> (x X. x) ~~ x)
48	47	anim2i 333	. . . . . . . . . . . . . . . . . . 19 |- ((om ~<_ x /\ g:(x X. x)-1-1-onto->x) -> (om ~<_ x /\ (x X. x) ~~ x))
49	48	adantlr 393	. . . . . . . . . . . . . . . . . 18 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (om ~<_ x /\ (x X. x) ~~ x))
50	45, 49	sylan 450	. . . . . . . . . . . . . . . . 17 |- ((((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) /\ x ~~ y) -> E.u u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y)
51	50	adantrr 395	. . . . . . . . . . . . . . . 16 |- ((((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y (_ (A \ x))) -> E.u u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y)
52	1, 2	infxpidmlem11 7774	. . . . . . . . . . . . . . . . . 18 |- (((((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y (_ (A \ x))) /\ u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y) -> E.h e. H g (. h)
53	52	ex 371	. . . . . . . . . . . . . . . . 17 |- ((((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y (_ (A \ x))) -> (u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y -> E.h e. H g (. h))
54	53	19.23adv 1251	. . . . . . . . . . . . . . . 16 |- ((((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y (_ (A \ x))) -> (E.u u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y -> E.h e. H g (. h))
55	51, 54	mpd 26	. . . . . . . . . . . . . . 15 |- ((((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y (_ (A \ x))) -> E.h e. H g (. h)
56	55	ex 371	. . . . . . . . . . . . . 14 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> ((x ~~ y /\ y (_ (A \ x)) -> E.h e. H g (. h))
57	56	19.23adv 1251	. . . . . . . . . . . . 13 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (E.y(x ~~ y /\ y (_ (A \ x)) -> E.h e. H g (. h))
58		difexg 2796	. . . . . . . . . . . . . . 15 |- (A e. V -> (A \ x) e. V)
59	2, 58	ax-mp 7	. . . . . . . . . . . . . 14 |- (A \ x) e. V
60	59	domen 4520	. . . . . . . . . . . . 13 |- (x ~<_ (A \ x) <-> E.y(x ~~ y /\ y (_ (A \ x)))
61	57, 60	syl5ib 204	. . . . . . . . . . . 12 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (x ~<_ (A \ x) -> E.h e. H g (. h))
62		domtri 4987	. . . . . . . . . . . . 13 |- ((x e. V /\ (A \ x) e. V) -> (x ~<_ (A \ x) <-> -. (A \ x) ~< x))
63	10, 59, 62	mp2an 701	. . . . . . . . . . . 12 |- (x ~<_ (A \ x) <-> -. (A \ x) ~< x)
64		dfrex2 1702	. . . . . . . . . . . 12 |- (E.h e. H g (. h <-> -. A.h e. H -. g (. h)
65	61, 63, 64	3imtr3g 555	. . . . . . . . . . 11 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (-. (A \ x) ~< x -> -. A.h e. H -. g (. h))
66	65	con4d 75	. . . . . . . . . 10 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (A.h e. H -. g (. h -> (A \ x) ~< x))
67		sbth 4602	. . . . . . . . . . . . . . . 16 |- ((A ~<_ x /\ x ~<_ A) -> A ~~ x)
68		domentr 4562	. . . . . . . . . . . . . . . . 17 |- ((A ~<_ (x X. x) /\ (x X. x) ~~ x) -> A ~<_ x)
69	10, 59	unxpdom2 4995	. . . . . . . . . . . . . . . . . . 19 |- ((1o ~< x /\ (A \ x) ~<_ x) -> (x u. (A \ x)) ~<_ (x X. x))
70		ssun2 2246	. . . . . . . . . . . . . . . . . . . . . 22 |- A (_ (x u. A)
71		ssdomg 4549	. . . . . . . . . . . . . . . . . . . . . 22 |- (A e. V -> (A (_ (x u. A) -> A ~<_ (x u. A)))
72	2, 70, 71	mp2 43	. . . . . . . . . . . . . . . . . . . . 21 |- A ~<_ (x u. A)
73		undif2 2394	. . . . . . . . . . . . . . . . . . . . 21 |- (x u. (A \ x)) = (x u. A)
74	72, 73	breqtrri 2713	. . . . . . . . . . . . . . . . . . . 20 |- A ~<_ (x u. (A \ x))
75		domtr 4556	. . . . . . . . . . . . . . . . . . . 20 |- ((A ~<_ (x u. (A \ x)) /\ (x u. (A \ x)) ~<_ (x X. x)) -> A ~<_ (x X. x))
76	74, 75	mpan 699	. . . . . . . . . . . . . . . . . . 19 |- ((x u. (A \ x)) ~<_ (x X. x) -> A ~<_ (x X. x))
77	69, 76	syl 10	. . . . . . . . . . . . . . . . . 18 |- ((1o ~< x /\ (A \ x) ~<_ x) -> A ~<_ (x X. x))
78		1onn 4393	. . . . . . . . . . . . . . . . . . 19 |- 1o e. om
79	10	infsdomnn 4678	. . . . . . . . . . . . . . . . . . 19 |- ((om ~<_ x /\ 1o e. om) -> 1o ~< x)
80	78, 79	mpan2 700	. . . . . . . . . . . . . . . . . 18 |- (om ~<_ x -> 1o ~< x)
81		sdomdom 4527	. . . . . . . . . . . . . . . . . 18 |- ((A \ x) ~< x -> (A \ x) ~<_ x)
82	77, 80, 81	syl2an 456	. . . . . . . . . . . . . . . . 17 |- ((om ~<_ x /\ (A \ x) ~< x) -> A ~<_ (x X. x))
83	68, 82	sylan 450	. . . . . . . . . . . . . . . 16 |- (((om ~<_ x /\ (A \ x) ~< x) /\ (x X. x) ~~ x) -> A ~<_ x)
84		ssdomg 4549	. . . . . . . . . . . . . . . . 17 |- (x e. V -> (x (_ A -> x ~<_ A))
85	10, 84	ax-mp 7	. . . . . . . . . . . . . . . 16 |- (x (_ A -> x ~<_ A)
86	67, 83, 85	syl2an 456	. . . . . . . . . . . . . . 15 |- ((((om ~<_ x /\ (A \ x) ~< x) /\ (x X. x) ~~ x) /\ x (_ A) -> A ~~ x)
87		entr 4555	. . . . . . . . . . . . . . . . . 18 |- (((A X. A) ~~ x /\ x ~~ A) -> (A X. A) ~~ A)
88		entr 4555	. . . . . . . . . . . . . . . . . . 19 |- (((A X. A) ~~ (x X. x) /\ (x X. x) ~~ x) -> (A X. A) ~~ x)
89	2, 10, 2, 10	xpen 4635	. . . . . . . . . . . . . . . . . . . 20 |- ((A ~~ x /\ A ~~ x) -> (A X. A) ~~ (x X. x))
90	89	anidms 435	. . . . . . . . . . . . . . . . . . 19 |- (A ~~ x -> (A X. A) ~~ (x X. x))
91	88, 90	sylan 450	. . . . . . . . . . . . . . . . . 18 |- ((A ~~ x /\ (x X. x) ~~ x) -> (A X. A) ~~ x)
92	10	ensym 4553	. . . . . . . . . . . . . . . . . . 19 |- (A ~~ x -> x ~~ A)
93	92	adantr 389	. . . . . . . . . . . . . . . . . 18 |- ((A ~~ x /\ (x X. x) ~~ x) -> x ~~ A)
94	87, 91, 93	sylanc 473	. . . . . . . . . . . . . . . . 17 |- ((A ~~ x /\ (x X. x) ~~ x) -> (A X. A) ~~ A)
95	94	expcom 372	. . . . . . . . . . . . . . . 16 |- ((x X. x) ~~ x -> (A ~~ x -> (A X. A) ~~ A))
96	95	ad2antlr 405	. . . . . . . . . . . . . . 15 |- ((((om ~<_ x /\ (A \ x) ~< x) /\ (x X. x) ~~ x) /\ x (_ A) -> (A ~~ x -> (A X. A) ~~ A))
97	86, 96	mpd 26	. . . . . . . . . . . . . 14 |- ((((om ~<_ x /\ (A \ x) ~< x) /\ (x X. x) ~~ x) /\ x (_ A) -> (A X. A) ~~ A)
98	97	exp41 382	. . . . . . . . . . . . 13 |- (om ~<_ x -> ((A \ x) ~< x -> ((x X. x) ~~ x -> (x (_ A -> (A X. A) ~~ A))))
99	98	com24 37	. . . . . . . . . . . 12 |- (om ~<_ x -> (x (_ A -> ((x X. x) ~~ x -> ((A \ x) ~< x -> (A X. A) ~~ A))))
100	99	imp31 360	. . . . . . . . . . 11 |- (((om ~<_ x /\ x (_ A) /\ (x X. x) ~~ x) -> ((A \ x) ~< x -> (A X. A) ~~ A))
101	100, 47	sylan2 453	. . . . . . . . . 10 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> ((A \ x) ~< x -> (A X. A) ~~ A))
102	66, 101	syld 27	. . . . . . . . 9 |- (((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (A.h e. H -. g (. h -> (A X. A) ~~ A))
103	102	19.23aiv 1333	. . . . . . . 8 |- (E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x) -> (A.h e. H -. g (. h -> (A X. A) ~~ A))
104	103	imim2i 17	. . . . . . 7 |- ((g =/= (/) -> E.x((om ~<_ x /\ x (_ A) /\ g:(x X. x)-1-1-onto->x)) -> (g =/= (/) -> (A.h e. H -. g (. h -> (A X. A) ~~ A)))
105	8, 104	sylbi 197	. . . . . 6 |- (g e. H -> (g =/= (/) -> (A.h e. H -. g (. h -> (A X. A) ~~ A)))
106	105	com13 33	. . . . 5 |- (A.h e. H -. g (. h -> (g =/= (/) -> (g e. H -> (A X. A) ~~ A)))
107	4, 106	syld 27	. . . 4 |- (A.h e. H -. g (. h -> (om ~<_ A -> (g e. H -> (A X. A) ~~ A)))
108	107	com3r 35	. . 3 |- (g e. H -> (A.h e. H -. g (. h -> (om ~<_ A -> (A X. A) ~~ A)))
109	108	r19.23aiv 1789	. 2 |- (E.g e. H A.h e. H -. g (. h -> (om ~<_ A -> (A X. A) ~~ A))
110	3, 109	ax-mp 7	1 |- (om ~<_ A -> (A X. A) ~~ A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994  E.wex 1016  {cab 1505   =/= wne 1628  A.wral 1691  E.wrex 1692  Vcvv 1857   \ cdif 2096   u. cun 2097   (_ wss 2099   (. wpss 2100  (/)c0 2332   class class class wbr 2692  omcom 3218   X. cxp 3249  -1-1-onto->wf1o 3262  1oc1o 4264   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507

This theorem is referenced by:  infxpidm 7776

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-iso 3280  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-2o 4270  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-fin 4512  df-card 4962  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-n 6070  df-2 6116  df-n0 6268  df-z 6304  df-seq1 6673  df-exp 6764

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