Theorem ltsosr 7831

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 7830	. 2 |- <R Po R.
2		df-nr 7794	. . . 4 |- R. = ( ( P. X. P. ) /. ~R )
3		breq1 4036	. . . . 5 |- ( [ <. a , b >. ] ~R = x -> ( [ <. a , b >. ] ~R <R [ <. c , d >. ] ~R <-> x <R [ <. c , d >. ] ~R ) )
4		breq1 4036	. . . . . 6 |- ( [ <. a , b >. ] ~R = x -> ( [ <. a , b >. ] ~R <R [ <. e , f >. ] ~R <-> x <R [ <. e , f >. ] ~R ) )
5	4	orbi1d 792	. . . . 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 4037	. . . . 5 |- ( [ <. c , d >. ] ~R = y -> ( x <R [ <. c , d >. ] ~R <-> x <R y ) )
8		breq2 4037	. . . . . 6 |- ( [ <. c , d >. ] ~R = y -> ( [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R <-> [ <. e , f >. ] ~R <R y ) )
9	8	orbi2d 791	. . . . 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 4037	. . . . . 6 |- ( [ <. e , f >. ] ~R = z -> ( x <R [ <. e , f >. ] ~R <-> x <R z ) )
12		breq1 4036	. . . . . 6 |- ( [ <. e , f >. ] ~R = z -> ( [ <. e , f >. ] ~R <R y <-> z <R y ) )
13	11, 12	orbi12d 794	. . . . 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 1023	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> a e. P. )
16		simp3r 1028	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> f e. P. )
17		addclpr 7604	. . . . . . . . 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 1026	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> d e. P. )
20		addclpr 7604	. . . . . . . 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 1025	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> c e. P. )
23		addclpr 7604	. . . . . . . . 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 1024	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> b e. P. )
26		addclpr 7604	. . . . . . . 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 1027	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> e e. P. )
29		addclpr 7604	. . . . . . . . 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 7604	. . . . . . . 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 7663	. . . . . . . 8 |- <P Or P.
34		sowlin 4355	. . . . . . . 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 1249	. . . . . 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 7604	. . . . . . . . 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 7604	. . . . . . . . 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 7686	. . . . . . . 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 1249	. . . . . . 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 7645	. . . . . . . . . . 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 7646	. . . . . . . . . . 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 6105	. . . . . . . . 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 6083	. . . . . . . . 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 2232	. . . . . . . 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 6083	. . . . . . . . 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 6104	. . . . . . . . 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 2231	. . . . . . . 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 4046	. . . . . . 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 7686	. . . . . . . . 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 6093	. . . . . . 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 7604	. . . . . . . . . 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 6093	. . . . . . . 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 6083	. . . . . . . . . 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 6080	. . . . . . . . . 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 2229	. . . . . . . . 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 4043	. . . . . . . 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 794	. . . . . 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 7814	. . . . . 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 1019	. . . . 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 7814	. . . . . . 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 1018	. . . . . 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 7814	. . . . . . . 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 1017	. . . . . 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 794	. . . . 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 6683	. . 3 |- ( ( x e. R. /\ y e. R. /\ z e. R. ) -> ( x <R y -> ( x <R z \/ z <R y ) ) )
78	77	rgen3 2584	. 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 4332	. 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 944	1 |- <R Or R.

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   <.cop 3625   class class class wbr 4033   Po wpo 4329   Or wor 4330  (class class class)co 5922   [cec 6590   P.cnp 7358   +P. cpp 7360   <P cltp 7362   ~R cer 7363   R.cnr 7364   <R cltr 7370

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-iplp 7535  df-iltp 7537  df-enr 7793  df-nr 7794  df-ltr 7797

This theorem is referenced by:  1ne0sr  7833  addgt0sr  7842  caucvgsrlemcl  7856  caucvgsrlemfv  7858  suplocsrlemb  7873  suplocsrlempr  7874  suplocsrlem  7875  axpre-ltirr  7949  axpre-ltwlin  7950  axpre-lttrn  7951