Theorem ltsosr 8027

Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)

Assertion

Ref	Expression
ltsosr	|- <R Or R.

Proof of Theorem ltsosr

Dummy variables a b c d e f r s t x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltposr 8026	. 2 |- <R Po R.
2		df-nr 7990	. . . 4 |- R. = ( ( P. X. P. ) /. ~R )
3		breq1 4096	. . . . 5 |- ( [ <. a , b >. ] ~R = x -> ( [ <. a , b >. ] ~R <R [ <. c , d >. ] ~R <-> x <R [ <. c , d >. ] ~R ) )
4		breq1 4096	. . . . . 6 |- ( [ <. a , b >. ] ~R = x -> ( [ <. a , b >. ] ~R <R [ <. e , f >. ] ~R <-> x <R [ <. e , f >. ] ~R ) )
5	4	orbi1d 799	. . . . 5 |- ( [ <. a , b >. ] ~R = x -> ( ( [ <. a , b >. ] ~R <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R ) <-> ( x <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R ) ) )
6	3, 5	imbi12d 234	. . . 4 |- ( [ <. a , b >. ] ~R = x -> ( ( [ <. a , b >. ] ~R <R [ <. c , d >. ] ~R -> ( [ <. a , b >. ] ~R <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R ) ) <-> ( x <R [ <. c , d >. ] ~R -> ( x <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R ) ) ) )
7		breq2 4097	. . . . 5 |- ( [ <. c , d >. ] ~R = y -> ( x <R [ <. c , d >. ] ~R <-> x <R y ) )
8		breq2 4097	. . . . . 6 |- ( [ <. c , d >. ] ~R = y -> ( [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R <-> [ <. e , f >. ] ~R <R y ) )
9	8	orbi2d 798	. . . . 5 |- ( [ <. c , d >. ] ~R = y -> ( ( x <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R ) <-> ( x <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R y ) ) )
10	7, 9	imbi12d 234	. . . 4 |- ( [ <. c , d >. ] ~R = y -> ( ( x <R [ <. c , d >. ] ~R -> ( x <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R ) ) <-> ( x <R y -> ( x <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R y ) ) ) )
11		breq2 4097	. . . . . 6 |- ( [ <. e , f >. ] ~R = z -> ( x <R [ <. e , f >. ] ~R <-> x <R z ) )
12		breq1 4096	. . . . . 6 |- ( [ <. e , f >. ] ~R = z -> ( [ <. e , f >. ] ~R <R y <-> z <R y ) )
13	11, 12	orbi12d 801	. . . . 5 |- ( [ <. e , f >. ] ~R = z -> ( ( x <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R y ) <-> ( x <R z \/ z <R y ) ) )
14	13	imbi2d 230	. . . 4 |- ( [ <. e , f >. ] ~R = z -> ( ( x <R y -> ( x <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R y ) ) <-> ( x <R y -> ( x <R z \/ z <R y ) ) ) )
15		simp1l 1048	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> a e. P. )
16		simp3r 1053	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> f e. P. )
17		addclpr 7800	. . . . . . . . 9 |- ( ( a e. P. /\ f e. P. ) -> ( a +P. f ) e. P. )
18	15, 16, 17	syl2anc 411	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( a +P. f ) e. P. )
19		simp2r 1051	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> d e. P. )
20		addclpr 7800	. . . . . . . 8 |- ( ( ( a +P. f ) e. P. /\ d e. P. ) -> ( ( a +P. f ) +P. d ) e. P. )
21	18, 19, 20	syl2anc 411	. . . . . . 7 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( a +P. f ) +P. d ) e. P. )
22		simp2l 1050	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> c e. P. )
23		addclpr 7800	. . . . . . . . 9 |- ( ( f e. P. /\ c e. P. ) -> ( f +P. c ) e. P. )
24	16, 22, 23	syl2anc 411	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( f +P. c ) e. P. )
25		simp1r 1049	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> b e. P. )
26		addclpr 7800	. . . . . . . 8 |- ( ( ( f +P. c ) e. P. /\ b e. P. ) -> ( ( f +P. c ) +P. b ) e. P. )
27	24, 25, 26	syl2anc 411	. . . . . . 7 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( f +P. c ) +P. b ) e. P. )
28		simp3l 1052	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> e e. P. )
29		addclpr 7800	. . . . . . . . 9 |- ( ( b e. P. /\ e e. P. ) -> ( b +P. e ) e. P. )
30	25, 28, 29	syl2anc 411	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( b +P. e ) e. P. )
31		addclpr 7800	. . . . . . . 8 |- ( ( ( b +P. e ) e. P. /\ d e. P. ) -> ( ( b +P. e ) +P. d ) e. P. )
32	30, 19, 31	syl2anc 411	. . . . . . 7 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( b +P. e ) +P. d ) e. P. )
33		ltsopr 7859	. . . . . . . 8 |- <P Or P.
34		sowlin 4423	. . . . . . . 8 |- ( ( <P Or P. /\ ( ( ( a +P. f ) +P. d ) e. P. /\ ( ( f +P. c ) +P. b ) e. P. /\ ( ( b +P. e ) +P. d ) e. P. ) ) -> ( ( ( a +P. f ) +P. d ) <P ( ( f +P. c ) +P. b ) -> ( ( ( a +P. f ) +P. d ) <P ( ( b +P. e ) +P. d ) \/ ( ( b +P. e ) +P. d ) <P ( ( f +P. c ) +P. b ) ) ) )
35	33, 34	mpan 424	. . . . . . 7 |- ( ( ( ( a +P. f ) +P. d ) e. P. /\ ( ( f +P. c ) +P. b ) e. P. /\ ( ( b +P. e ) +P. d ) e. P. ) -> ( ( ( a +P. f ) +P. d ) <P ( ( f +P. c ) +P. b ) -> ( ( ( a +P. f ) +P. d ) <P ( ( b +P. e ) +P. d ) \/ ( ( b +P. e ) +P. d ) <P ( ( f +P. c ) +P. b ) ) ) )
36	21, 27, 32, 35	syl3anc 1274	. . . . . 6 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( ( a +P. f ) +P. d ) <P ( ( f +P. c ) +P. b ) -> ( ( ( a +P. f ) +P. d ) <P ( ( b +P. e ) +P. d ) \/ ( ( b +P. e ) +P. d ) <P ( ( f +P. c ) +P. b ) ) ) )
37		addclpr 7800	. . . . . . . . 9 |- ( ( a e. P. /\ d e. P. ) -> ( a +P. d ) e. P. )
38	15, 19, 37	syl2anc 411	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( a +P. d ) e. P. )
39		addclpr 7800	. . . . . . . . 9 |- ( ( b e. P. /\ c e. P. ) -> ( b +P. c ) e. P. )
40	25, 22, 39	syl2anc 411	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( b +P. c ) e. P. )
41		ltaprg 7882	. . . . . . . 8 |- ( ( ( a +P. d ) e. P. /\ ( b +P. c ) e. P. /\ f e. P. ) -> ( ( a +P. d ) <P ( b +P. c ) <-> ( f +P. ( a +P. d ) ) <P ( f +P. ( b +P. c ) ) ) )
42	38, 40, 16, 41	syl3anc 1274	. . . . . . 7 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( a +P. d ) <P ( b +P. c ) <-> ( f +P. ( a +P. d ) ) <P ( f +P. ( b +P. c ) ) ) )
43		addcomprg 7841	. . . . . . . . . . 11 |- ( ( r e. P. /\ s e. P. ) -> ( r +P. s ) = ( s +P. r ) )
44	43	adantl 277	. . . . . . . . . 10 |- ( ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) /\ ( r e. P. /\ s e. P. ) ) -> ( r +P. s ) = ( s +P. r ) )
45		addassprg 7842	. . . . . . . . . . 11 |- ( ( r e. P. /\ s e. P. /\ t e. P. ) -> ( ( r +P. s ) +P. t ) = ( r +P. ( s +P. t ) ) )
46	45	adantl 277	. . . . . . . . . 10 |- ( ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) /\ ( r e. P. /\ s e. P. /\ t e. P. ) ) -> ( ( r +P. s ) +P. t ) = ( r +P. ( s +P. t ) ) )
47	16, 15, 19, 44, 46	caov12d 6214	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( f +P. ( a +P. d ) ) = ( a +P. ( f +P. d ) ) )
48	46, 15, 16, 19	caovassd 6192	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( a +P. f ) +P. d ) = ( a +P. ( f +P. d ) ) )
49	47, 48	eqtr4d 2267	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( f +P. ( a +P. d ) ) = ( ( a +P. f ) +P. d ) )
50	46, 16, 25, 22	caovassd 6192	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( f +P. b ) +P. c ) = ( f +P. ( b +P. c ) ) )
51	16, 25, 22, 44, 46	caov32d 6213	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( f +P. b ) +P. c ) = ( ( f +P. c ) +P. b ) )
52	50, 51	eqtr3d 2266	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( f +P. ( b +P. c ) ) = ( ( f +P. c ) +P. b ) )
53	49, 52	breq12d 4106	. . . . . . 7 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( f +P. ( a +P. d ) ) <P ( f +P. ( b +P. c ) ) <-> ( ( a +P. f ) +P. d ) <P ( ( f +P. c ) +P. b ) ) )
54	42, 53	bitrd 188	. . . . . 6 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( a +P. d ) <P ( b +P. c ) <-> ( ( a +P. f ) +P. d ) <P ( ( f +P. c ) +P. b ) ) )
55		ltaprg 7882	. . . . . . . . 9 |- ( ( r e. P. /\ s e. P. /\ t e. P. ) -> ( r <P s <-> ( t +P. r ) <P ( t +P. s ) ) )
56	55	adantl 277	. . . . . . . 8 |- ( ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) /\ ( r e. P. /\ s e. P. /\ t e. P. ) ) -> ( r <P s <-> ( t +P. r ) <P ( t +P. s ) ) )
57	56, 18, 30, 19, 44	caovord2d 6202	. . . . . . 7 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( a +P. f ) <P ( b +P. e ) <-> ( ( a +P. f ) +P. d ) <P ( ( b +P. e ) +P. d ) ) )
58		addclpr 7800	. . . . . . . . . 10 |- ( ( e e. P. /\ d e. P. ) -> ( e +P. d ) e. P. )
59	28, 19, 58	syl2anc 411	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( e +P. d ) e. P. )
60	56, 59, 24, 25, 44	caovord2d 6202	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( e +P. d ) <P ( f +P. c ) <-> ( ( e +P. d ) +P. b ) <P ( ( f +P. c ) +P. b ) ) )
61	46, 25, 28, 19	caovassd 6192	. . . . . . . . . 10 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( b +P. e ) +P. d ) = ( b +P. ( e +P. d ) ) )
62	44, 25, 59	caovcomd 6189	. . . . . . . . . 10 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( b +P. ( e +P. d ) ) = ( ( e +P. d ) +P. b ) )
63	61, 62	eqtrd 2264	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( b +P. e ) +P. d ) = ( ( e +P. d ) +P. b ) )
64	63	breq1d 4103	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( ( b +P. e ) +P. d ) <P ( ( f +P. c ) +P. b ) <-> ( ( e +P. d ) +P. b ) <P ( ( f +P. c ) +P. b ) ) )
65	60, 64	bitr4d 191	. . . . . . 7 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( e +P. d ) <P ( f +P. c ) <-> ( ( b +P. e ) +P. d ) <P ( ( f +P. c ) +P. b ) ) )
66	57, 65	orbi12d 801	. . . . . 6 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( ( a +P. f ) <P ( b +P. e ) \/ ( e +P. d ) <P ( f +P. c ) ) <-> ( ( ( a +P. f ) +P. d ) <P ( ( b +P. e ) +P. d ) \/ ( ( b +P. e ) +P. d ) <P ( ( f +P. c ) +P. b ) ) ) )
67	36, 54, 66	3imtr4d 203	. . . . 5 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( a +P. d ) <P ( b +P. c ) -> ( ( a +P. f ) <P ( b +P. e ) \/ ( e +P. d ) <P ( f +P. c ) ) ) )
68		ltsrprg 8010	. . . . . 6 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) ) -> ( [ <. a , b >. ] ~R <R [ <. c , d >. ] ~R <-> ( a +P. d ) <P ( b +P. c ) ) )
69	68	3adant3 1044	. . . . 5 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( [ <. a , b >. ] ~R <R [ <. c , d >. ] ~R <-> ( a +P. d ) <P ( b +P. c ) ) )
70		ltsrprg 8010	. . . . . . 7 |- ( ( ( a e. P. /\ b e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( [ <. a , b >. ] ~R <R [ <. e , f >. ] ~R <-> ( a +P. f ) <P ( b +P. e ) ) )
71	70	3adant2 1043	. . . . . 6 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( [ <. a , b >. ] ~R <R [ <. e , f >. ] ~R <-> ( a +P. f ) <P ( b +P. e ) ) )
72		ltsrprg 8010	. . . . . . . 8 |- ( ( ( e e. P. /\ f e. P. ) /\ ( c e. P. /\ d e. P. ) ) -> ( [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R <-> ( e +P. d ) <P ( f +P. c ) ) )
73	72	ancoms 268	. . . . . . 7 |- ( ( ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R <-> ( e +P. d ) <P ( f +P. c ) ) )
74	73	3adant1 1042	. . . . . 6 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R <-> ( e +P. d ) <P ( f +P. c ) ) )
75	71, 74	orbi12d 801	. . . . 5 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( ( [ <. a , b >. ] ~R <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R ) <-> ( ( a +P. f ) <P ( b +P. e ) \/ ( e +P. d ) <P ( f +P. c ) ) ) )
76	67, 69, 75	3imtr4d 203	. . . 4 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> ( [ <. a , b >. ] ~R <R [ <. c , d >. ] ~R -> ( [ <. a , b >. ] ~R <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R ) ) )
77	2, 6, 10, 14, 76	3ecoptocl 6836	. . 3 |- ( ( x e. R. /\ y e. R. /\ z e. R. ) -> ( x <R y -> ( x <R z \/ z <R y ) ) )
78	77	rgen3 2620	. 2 |- A. x e. R. A. y e. R. A. z e. R. ( x <R y -> ( x <R z \/ z <R y ) )
79		df-iso 4400	. 2 |- ( <R Or R. <-> ( <R Po R. /\ A. x e. R. A. y e. R. A. z e. R. ( x <R y -> ( x <R z \/ z <R y ) ) ) )
80	1, 78, 79	mpbir2an 951	1 |- <R Or R.

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   <.cop 3676   class class class wbr 4093   Po wpo 4397   Or wor 4398  (class class class)co 6028   [cec 6743   P.cnp 7554   +P. cpp 7556   <P cltp 7558   ~R cer 7559   R.cnr 7560   <R cltr 7566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-iplp 7731  df-iltp 7733  df-enr 7989  df-nr 7990  df-ltr 7993

This theorem is referenced by:  1ne0sr  8029  addgt0sr  8038  caucvgsrlemcl  8052  caucvgsrlemfv  8054  suplocsrlemb  8069  suplocsrlempr  8070  suplocsrlem  8071  axpre-ltirr  8145  axpre-ltwlin  8146  axpre-lttrn  8147