Theorem ltsosr 7848

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 7847	. 2 |- <R Po R.
2		df-nr 7811	. . . 4 |- R. = ( ( P. X. P. ) /. ~R )
3		breq1 4037	. . . . 5 |- ( [ <. a , b >. ] ~R = x -> ( [ <. a , b >. ] ~R <R [ <. c , d >. ] ~R <-> x <R [ <. c , d >. ] ~R ) )
4		breq1 4037	. . . . . 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 4038	. . . . 5 |- ( [ <. c , d >. ] ~R = y -> ( x <R [ <. c , d >. ] ~R <-> x <R y ) )
8		breq2 4038	. . . . . 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 4038	. . . . . 6 |- ( [ <. e , f >. ] ~R = z -> ( x <R [ <. e , f >. ] ~R <-> x <R z ) )
12		breq1 4037	. . . . . 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 7621	. . . . . . . . 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 7621	. . . . . . . 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 7621	. . . . . . . . 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 7621	. . . . . . . 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 7621	. . . . . . . . 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 7621	. . . . . . . 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 7680	. . . . . . . 8 |- <P Or P.
34		sowlin 4356	. . . . . . . 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 7621	. . . . . . . . 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 7621	. . . . . . . . 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 7703	. . . . . . . 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 7662	. . . . . . . . . . 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 7663	. . . . . . . . . . 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 6109	. . . . . . . . 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 6087	. . . . . . . . 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 6087	. . . . . . . . 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 6108	. . . . . . . . 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 4047	. . . . . . 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 7703	. . . . . . . . 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 6097	. . . . . . 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 7621	. . . . . . . . . 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 6097	. . . . . . . 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 6087	. . . . . . . . . 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 6084	. . . . . . . . . 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 4044	. . . . . . . 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 7831	. . . . . 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 7831	. . . . . . 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 7831	. . . . . . . 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 6692	. . 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 4333	. 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 3626   class class class wbr 4034   Po wpo 4330   Or wor 4331  (class class class)co 5925   [cec 6599   P.cnp 7375   +P. cpp 7377   <P cltp 7379   ~R cer 7380   R.cnr 7381   <R cltr 7387

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-iltp 7554  df-enr 7810  df-nr 7811  df-ltr 7814

This theorem is referenced by:  1ne0sr  7850  addgt0sr  7859  caucvgsrlemcl  7873  caucvgsrlemfv  7875  suplocsrlemb  7890  suplocsrlempr  7891  suplocsrlem  7892  axpre-ltirr  7966  axpre-ltwlin  7967  axpre-lttrn  7968