Theorem ltsosr 6849

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 6848	. 2 |- <R Po R.
2		df-nr 6812	. . . 4 |- R. = ( ( P. X. P. ) /. ~R )
3		breq1 3767	. . . . 5 |- ( [ <. a , b >. ] ~R = x -> ( [ <. a , b >. ] ~R <R [ <. c , d >. ] ~R <-> x <R [ <. c , d >. ] ~R ) )
4		breq1 3767	. . . . . 6 |- ( [ <. a , b >. ] ~R = x -> ( [ <. a , b >. ] ~R <R [ <. e , f >. ] ~R <-> x <R [ <. e , f >. ] ~R ) )
5	4	orbi1d 705	. . . . 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 223	. . . 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 3768	. . . . 5 |- ( [ <. c , d >. ] ~R = y -> ( x <R [ <. c , d >. ] ~R <-> x <R y ) )
8		breq2 3768	. . . . . 6 |- ( [ <. c , d >. ] ~R = y -> ( [ <. e , f >. ] ~R <R [ <. c , d >. ] ~R <-> [ <. e , f >. ] ~R <R y ) )
9	8	orbi2d 704	. . . . 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 223	. . . 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 3768	. . . . . 6 |- ( [ <. e , f >. ] ~R = z -> ( x <R [ <. e , f >. ] ~R <-> x <R z ) )
12		breq1 3767	. . . . . 6 |- ( [ <. e , f >. ] ~R = z -> ( [ <. e , f >. ] ~R <R y <-> z <R y ) )
13	11, 12	orbi12d 707	. . . . 5 |- ( [ <. e , f >. ] ~R = z -> ( ( x <R [ <. e , f >. ] ~R \/ [ <. e , f >. ] ~R <R y ) <-> ( x <R z \/ z <R y ) ) )
14	13	imbi2d 219	. . . 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 928	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> a e. P. )
16		simp3r 933	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> f e. P. )
17		addclpr 6635	. . . . . . . . 9 |- ( ( a e. P. /\ f e. P. ) -> ( a +P. f ) e. P. )
18	15, 16, 17	syl2anc 391	. . . . . . . 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 931	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> d e. P. )
20		addclpr 6635	. . . . . . . 8 |- ( ( ( a +P. f ) e. P. /\ d e. P. ) -> ( ( a +P. f ) +P. d ) e. P. )
21	18, 19, 20	syl2anc 391	. . . . . . 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 930	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> c e. P. )
23		addclpr 6635	. . . . . . . . 9 |- ( ( f e. P. /\ c e. P. ) -> ( f +P. c ) e. P. )
24	16, 22, 23	syl2anc 391	. . . . . . . 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 929	. . . . . . . 8 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> b e. P. )
26		addclpr 6635	. . . . . . . 8 |- ( ( ( f +P. c ) e. P. /\ b e. P. ) -> ( ( f +P. c ) +P. b ) e. P. )
27	24, 25, 26	syl2anc 391	. . . . . . 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 932	. . . . . . . . 9 |- ( ( ( a e. P. /\ b e. P. ) /\ ( c e. P. /\ d e. P. ) /\ ( e e. P. /\ f e. P. ) ) -> e e. P. )
29		addclpr 6635	. . . . . . . . 9 |- ( ( b e. P. /\ e e. P. ) -> ( b +P. e ) e. P. )
30	25, 28, 29	syl2anc 391	. . . . . . . 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 6635	. . . . . . . 8 |- ( ( ( b +P. e ) e. P. /\ d e. P. ) -> ( ( b +P. e ) +P. d ) e. P. )
32	30, 19, 31	syl2anc 391	. . . . . . 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 6694	. . . . . . . 8 |- <P Or P.
34		sowlin 4057	. . . . . . . 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 400	. . . . . . 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 1135	. . . . . 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 6635	. . . . . . . . 9 |- ( ( a e. P. /\ d e. P. ) -> ( a +P. d ) e. P. )
38	15, 19, 37	syl2anc 391	. . . . . . . 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 6635	. . . . . . . . 9 |- ( ( b e. P. /\ c e. P. ) -> ( b +P. c ) e. P. )
40	25, 22, 39	syl2anc 391	. . . . . . . 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 6717	. . . . . . . 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 1135	. . . . . . 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 6676	. . . . . . . . . . 11 |- ( ( r e. P. /\ s e. P. ) -> ( r +P. s ) = ( s +P. r ) )
44	43	adantl 262	. . . . . . . . . 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 6677	. . . . . . . . . . 11 |- ( ( r e. P. /\ s e. P. /\ t e. P. ) -> ( ( r +P. s ) +P. t ) = ( r +P. ( s +P. t ) ) )
46	45	adantl 262	. . . . . . . . . 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 5682	. . . . . . . . 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 5660	. . . . . . . . 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 2075	. . . . . . . 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 5660	. . . . . . . . 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 5681	. . . . . . . . 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 2074	. . . . . . . 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 3777	. . . . . . 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 177	. . . . . 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 6717	. . . . . . . . 9 |- ( ( r e. P. /\ s e. P. /\ t e. P. ) -> ( r <P s <-> ( t +P. r ) <P ( t +P. s ) ) )
56	55	adantl 262	. . . . . . . 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 5670	. . . . . . 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 6635	. . . . . . . . . 10 |- ( ( e e. P. /\ d e. P. ) -> ( e +P. d ) e. P. )
59	28, 19, 58	syl2anc 391	. . . . . . . . 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 5670	. . . . . . . 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 5660	. . . . . . . . . 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 5657	. . . . . . . . . 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 2072	. . . . . . . . 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 3774	. . . . . . . 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 180	. . . . . . 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 707	. . . . . 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 192	. . . . 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 6832	. . . . . 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 924	. . . . 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 6832	. . . . . . 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 923	. . . . . 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 6832	. . . . . . . 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 255	. . . . . . 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 922	. . . . . 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 707	. . . . 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 192	. . . 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 6195	. . 3 |- ( ( x e. R. /\ y e. R. /\ z e. R. ) -> ( x <R y -> ( x <R z \/ z <R y ) ) )
78	77	rgen3 2406	. 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 4034	. 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 849	1 |- <R Or R.

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   /\ w3a 885   = wceq 1243   e. wcel 1393   A.wral 2306   <.cop 3378   class class class wbr 3764   Po wpo 4031   Or wor 4032  (class class class)co 5512   [cec 6104   P.cnp 6389   +P. cpp 6391   <P cltp 6393   ~R cer 6394   R.cnr 6395   <R cltr 6401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815

This theorem is referenced by:  1ne0sr  6851  addgt0sr  6860  caucvgsrlemcl  6873  caucvgsrlemfv  6875  axpre-ltirr  6956  axpre-ltwlin  6957  axpre-lttrn  6958