Theorem infxpidmlem12 8832

Description: Lemma for infxpidm 8833. Letting x be the range of a maximal bijection g in H, infxpidmlem11 8831 lets us show that assuming x ~<_ (A \ x) leads to the contradiction E.h e. Hg C. 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 8829, A is also idempotent.

Hypotheses

Ref	Expression
infxpidmlem.1	|- H = {f | (f = (/) \/ E.t((om ~<_ t /\ t C_ 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 C_ A) /\ f:(t X. t)-1-1-onto->t))}
2		infxpidmlem.2	. . 3 |- A e. _V
3	1, 2	infxpidmlem9 8829	. 2 |- E.g e. H A.h e. H -. g C. h
4	1, 2	infxpidmlem10 8830	. . . . 5 |- (A.h e. H -. g C. h -> (om ~<_ A -> g =/= (/)))
5		visset 2295	. . . . . . . . 9 |- g e. _V
6	1, 5	infxpidmlem2 8822	. . . . . . . 8 |- (g e. H <-> (g = (/) \/ E.x((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x)))
7		neor 2096	. . . . . . . 8 |- ((g = (/) \/ E.x((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x)) <-> (g =/= (/) -> E.x((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x)))
8	6, 7	bitri 190	. . . . . . 7 |- (g e. H <-> (g =/= (/) -> E.x((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x)))
9		entr 5473	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (x X. y)) -> x ~~ (x X. y))
10		visset 2295	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- x e. _V
11	10	enref 5450	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- x ~~ x
12		visset 2295	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- y e. _V
13	10, 10, 10, 12	xpen 5582	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((x ~~ x /\ x ~~ y) -> (x X. x) ~~ (x X. y))
14	11, 13	mpan 759	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x ~~ y -> (x X. x) ~~ (x X. y))
15	9, 14	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((x ~~ (x X. x) /\ x ~~ y) -> x ~~ (x X. y))
16	15	adantll 428	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> x ~~ (x X. y))
17		entr 5473	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (y X. x)) -> x ~~ (y X. x))
18	10, 12, 10, 10	xpen 5582	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((x ~~ y /\ x ~~ x) -> (x X. x) ~~ (y X. x))
19	11, 18	mpan2 760	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x ~~ y -> (x X. x) ~~ (y X. x))
20	17, 19	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((x ~~ (x X. x) /\ x ~~ y) -> x ~~ (y X. x))
21		entr 5473	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((x ~~ (x X. x) /\ (x X. x) ~~ (y X. y)) -> x ~~ (y X. y))
22	10, 12, 10, 12	xpen 5582	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((x ~~ y /\ x ~~ y) -> (x X. x) ~~ (y X. y))
23	22	anidms 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x ~~ y -> (x X. x) ~~ (y X. y))
24	21, 23	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((x ~~ (x X. x) /\ x ~~ y) -> x ~~ (y X. y))
25	20, 24	jca 310	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((x ~~ (x X. x) /\ x ~~ y) -> (x ~~ (y X. x) /\ x ~~ (y X. y)))
26	25	adantll 428	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> (x ~~ (y X. x) /\ x ~~ (y X. y)))
27	12, 10	xpex 4096	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y X. x) e. _V
28	12, 12	xpex 4096	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y X. y) e. _V
29	27, 28	infxpidmlem1 8821	. . . . . . . . . . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . . . . . . . . . . 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 29	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> x ~~ ((y X. x) u. (y X. y)))
32	10, 12	xpex 4096	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x X. y) e. _V
33	27, 28	unex 3796	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((y X. x) u. (y X. y)) e. _V
34	32, 33	infxpidmlem1 8821	. . . . . . . . . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . 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 3796	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((x X. y) u. ((y X. x) u. (y X. y))) e. _V
38	37	ensym 5471	. . . . . . . . . . . . . . . . . . . . . 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 12	. . . . . . . . . . . . . . . . . . . . 21 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> ((x X. y) u. ((y X. x) u. (y X. y))) ~~ x)
40		entr 5473	. . . . . . . . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . . . . . . . 20 |- (((om ~<_ x /\ x ~~ (x X. x)) /\ x ~~ y) -> ((x X. y) u. ((y X. x) u. (y X. y))) ~~ y)
42	10	ensym 5471	. . . . . . . . . . . . . . . . . . . 20 |- ((x X. x) ~~ x -> x ~~ (x X. x))
43	41, 42	sylanl2 510	. . . . . . . . . . . . . . . . . . 19 |- (((om ~<_ x /\ (x X. x) ~~ x) /\ x ~~ y) -> ((x X. y) u. ((y X. x) u. (y X. y))) ~~ y)
44	12	bren 5436	. . . . . . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . . . . . . 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 4096	. . . . . . . . . . . . . . . . . . . . 21 |- (x X. x) e. _V
47	46	f1oen 5457	. . . . . . . . . . . . . . . . . . . 20 |- (g:(x X. x)-1-1-onto->x -> (x X. x) ~~ x)
48	47	anim2i 362	. . . . . . . . . . . . . . . . . . 19 |- ((om ~<_ x /\ g:(x X. x)-1-1-onto->x) -> (om ~<_ x /\ (x X. x) ~~ x))
49	48	adantlr 429	. . . . . . . . . . . . . . . . . 18 |- (((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) -> (om ~<_ x /\ (x X. x) ~~ x))
50	45, 49	sylan 497	. . . . . . . . . . . . . . . . 17 |- ((((om ~<_ x /\ x C_ 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 431	. . . . . . . . . . . . . . . 16 |- ((((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y C_ (A \ x))) -> E.u u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y)
52	1, 2	infxpidmlem11 8831	. . . . . . . . . . . . . . . . . 18 |- (((((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y C_ (A \ x))) /\ u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y) -> E.h e. H g C. h)
53	52	ex 402	. . . . . . . . . . . . . . . . 17 |- ((((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y C_ (A \ x))) -> (u:((x X. y) u. ((y X. x) u. (y X. y)))-1-1-onto->y -> E.h e. H g C. h))
54	53	19.23adv 1584	. . . . . . . . . . . . . . . 16 |- ((((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y C_ (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 C. h))
55	51, 54	mpd 29	. . . . . . . . . . . . . . 15 |- ((((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) /\ (x ~~ y /\ y C_ (A \ x))) -> E.h e. H g C. h)
56	55	ex 402	. . . . . . . . . . . . . 14 |- (((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) -> ((x ~~ y /\ y C_ (A \ x)) -> E.h e. H g C. h))
57	56	19.23adv 1584	. . . . . . . . . . . . 13 |- (((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) -> (E.y(x ~~ y /\ y C_ (A \ x)) -> E.h e. H g C. h))
58		difexg 3458	. . . . . . . . . . . . . . 15 |- (A e. _V -> (A \ x) e. _V)
59	2, 58	ax-mp 7	. . . . . . . . . . . . . 14 |- (A \ x) e. _V
60	59	domen 5438	. . . . . . . . . . . . 13 |- (x ~<_ (A \ x) <-> E.y(x ~~ y /\ y C_ (A \ x)))
61	57, 60	syl5ib 223	. . . . . . . . . . . 12 |- (((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) -> (x ~<_ (A \ x) -> E.h e. H g C. h))
62		domtri 5989	. . . . . . . . . . . . 13 |- ((x e. _V /\ (A \ x) e. _V) -> (x ~<_ (A \ x) <-> -. (A \ x) ~< x))
63	10, 59, 62	mp2an 761	. . . . . . . . . . . 12 |- (x ~<_ (A \ x) <-> -. (A \ x) ~< x)
64		dfrex2 2116	. . . . . . . . . . . 12 |- (E.h e. H g C. h <-> -. A.h e. H -. g C. h)
65	61, 63, 64	3imtr3g 611	. . . . . . . . . . 11 |- (((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) -> (-. (A \ x) ~< x -> -. A.h e. H -. g C. h))
66	65	con4d 91	. . . . . . . . . 10 |- (((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) -> (A.h e. H -. g C. h -> (A \ x) ~< x))
67		sbth 5520	. . . . . . . . . . . . . . . 16 |- ((A ~<_ x /\ x ~<_ A) -> A ~~ x)
68		domentr 5480	. . . . . . . . . . . . . . . . 17 |- ((A ~<_ (x X. x) /\ (x X. x) ~~ x) -> A ~<_ x)
69		domtr 5474	. . . . . . . . . . . . . . . . . . 19 |- ((A ~<_ (x u. (A \ x)) /\ (x u. (A \ x)) ~<_ (x X. x)) -> A ~<_ (x X. x))
70		ssun2 2768	. . . . . . . . . . . . . . . . . . . . 21 |- A C_ (x u. A)
71		ssdomg 5467	. . . . . . . . . . . . . . . . . . . . 21 |- (A e. _V -> (A C_ (x u. A) -> A ~<_ (x u. A)))
72	2, 70, 71	mp2 54	. . . . . . . . . . . . . . . . . . . 20 |- A ~<_ (x u. A)
73		undif2 2950	. . . . . . . . . . . . . . . . . . . 20 |- (x u. (A \ x)) = (x u. A)
74	72, 73	breqtrri 3362	. . . . . . . . . . . . . . . . . . 19 |- A ~<_ (x u. (A \ x))
75	10, 59	unxpdom2 5997	. . . . . . . . . . . . . . . . . . 19 |- ((1o ~< x /\ (A \ x) ~<_ x) -> (x u. (A \ x)) ~<_ (x X. x))
76	69, 74, 75	sylancr 526	. . . . . . . . . . . . . . . . . 18 |- ((1o ~< x /\ (A \ x) ~<_ x) -> A ~<_ (x X. x))
77		1onn 5310	. . . . . . . . . . . . . . . . . . 19 |- 1o e. om
78	10	infsdomnn 5625	. . . . . . . . . . . . . . . . . . 19 |- ((om ~<_ x /\ 1o e. om) -> 1o ~< x)
79	77, 78	mpan2 760	. . . . . . . . . . . . . . . . . 18 |- (om ~<_ x -> 1o ~< x)
80		sdomdom 5445	. . . . . . . . . . . . . . . . . 18 |- ((A \ x) ~< x -> (A \ x) ~<_ x)
81	76, 79, 80	syl2an 503	. . . . . . . . . . . . . . . . 17 |- ((om ~<_ x /\ (A \ x) ~< x) -> A ~<_ (x X. x))
82	68, 81	sylan 497	. . . . . . . . . . . . . . . 16 |- (((om ~<_ x /\ (A \ x) ~< x) /\ (x X. x) ~~ x) -> A ~<_ x)
83		ssdomg 5467	. . . . . . . . . . . . . . . . 17 |- (x e. _V -> (x C_ A -> x ~<_ A))
84	10, 83	ax-mp 7	. . . . . . . . . . . . . . . 16 |- (x C_ A -> x ~<_ A)
85	67, 82, 84	syl2an 503	. . . . . . . . . . . . . . 15 |- ((((om ~<_ x /\ (A \ x) ~< x) /\ (x X. x) ~~ x) /\ x C_ A) -> A ~~ x)
86		entr 5473	. . . . . . . . . . . . . . . . . . 19 |- (((A X. A) ~~ (x X. x) /\ (x X. x) ~~ x) -> (A X. A) ~~ x)
87	2, 10, 2, 10	xpen 5582	. . . . . . . . . . . . . . . . . . . 20 |- ((A ~~ x /\ A ~~ x) -> (A X. A) ~~ (x X. x))
88	87	anidms 480	. . . . . . . . . . . . . . . . . . 19 |- (A ~~ x -> (A X. A) ~~ (x X. x))
89	86, 88	sylan 497	. . . . . . . . . . . . . . . . . 18 |- ((A ~~ x /\ (x X. x) ~~ x) -> (A X. A) ~~ x)
90	10	ensym 5471	. . . . . . . . . . . . . . . . . . 19 |- (A ~~ x -> x ~~ A)
91	90	adantr 425	. . . . . . . . . . . . . . . . . 18 |- ((A ~~ x /\ (x X. x) ~~ x) -> x ~~ A)
92		entr 5473	. . . . . . . . . . . . . . . . . 18 |- (((A X. A) ~~ x /\ x ~~ A) -> (A X. A) ~~ A)
93	89, 91, 92	syl11anc 524	. . . . . . . . . . . . . . . . 17 |- ((A ~~ x /\ (x X. x) ~~ x) -> (A X. A) ~~ A)
94	93	expcom 403	. . . . . . . . . . . . . . . 16 |- ((x X. x) ~~ x -> (A ~~ x -> (A X. A) ~~ A))
95	94	ad2antlr 441	. . . . . . . . . . . . . . 15 |- ((((om ~<_ x /\ (A \ x) ~< x) /\ (x X. x) ~~ x) /\ x C_ A) -> (A ~~ x -> (A X. A) ~~ A))
96	85, 95	mpd 29	. . . . . . . . . . . . . 14 |- ((((om ~<_ x /\ (A \ x) ~< x) /\ (x X. x) ~~ x) /\ x C_ A) -> (A X. A) ~~ A)
97	96	exp41 413	. . . . . . . . . . . . 13 |- (om ~<_ x -> ((A \ x) ~< x -> ((x X. x) ~~ x -> (x C_ A -> (A X. A) ~~ A))))
98	97	com24 41	. . . . . . . . . . . 12 |- (om ~<_ x -> (x C_ A -> ((x X. x) ~~ x -> ((A \ x) ~< x -> (A X. A) ~~ A))))
99	98	imp31 389	. . . . . . . . . . 11 |- (((om ~<_ x /\ x C_ A) /\ (x X. x) ~~ x) -> ((A \ x) ~< x -> (A X. A) ~~ A))
100	99, 47	sylan2 500	. . . . . . . . . 10 |- (((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) -> ((A \ x) ~< x -> (A X. A) ~~ A))
101	66, 100	syld 30	. . . . . . . . 9 |- (((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) -> (A.h e. H -. g C. h -> (A X. A) ~~ A))
102	101	19.23aiv 1674	. . . . . . . 8 |- (E.x((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x) -> (A.h e. H -. g C. h -> (A X. A) ~~ A))
103	102	imim2i 11	. . . . . . 7 |- ((g =/= (/) -> E.x((om ~<_ x /\ x C_ A) /\ g:(x X. x)-1-1-onto->x)) -> (g =/= (/) -> (A.h e. H -. g C. h -> (A X. A) ~~ A)))
104	8, 103	sylbi 216	. . . . . 6 |- (g e. H -> (g =/= (/) -> (A.h e. H -. g C. h -> (A X. A) ~~ A)))
105	104	com13 37	. . . . 5 |- (A.h e. H -. g C. h -> (g =/= (/) -> (g e. H -> (A X. A) ~~ A)))
106	4, 105	syld 30	. . . 4 |- (A.h e. H -. g C. h -> (om ~<_ A -> (g e. H -> (A X. A) ~~ A)))
107	106	com3r 39	. . 3 |- (g e. H -> (A.h e. H -. g C. h -> (om ~<_ A -> (A X. A) ~~ A)))
108	107	r19.23aiv 2211	. 2 |- (E.g e. H A.h e. H -. g C. h -> (om ~<_ A -> (A X. A) ~~ A))
109	3, 108	ax-mp 7	1 |- (om ~<_ A -> (A X. A) ~~ A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   \ cdif 2590   u. cun 2591   C_ wss 2593   C. wpss 2594  (/)c0 2875   class class class wbr 3338  omcom 3949   X. cxp 3984  -1-1-onto->wf1o 3997  1oc1o 5172   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425

This theorem is referenced by:  infxpidm 8833

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-inf2 5731  ax-ac 5906

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-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-iso 4015  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-2o 5178  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-undef 5556  df-riota 5560  df-card 5862  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

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