ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  lttrsr Unicode version

Theorem lttrsr 7538
Description: Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
Assertion
Ref Expression
lttrsr  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
( f  <R  g  /\  g  <R  h )  ->  f  <R  h
) )
Distinct variable group:    f, g, h

Proof of Theorem lttrsr
Dummy variables  r  s  t  x  y  z  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7503 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 3902 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  f  <R  [
<. z ,  w >. ]  ~R  ) )
32anbi1d 460 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  <->  ( f  <R  [ <. z ,  w >. ]  ~R  /\  [
<. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )
) )
4 breq1 3902 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  f  <R  [
<. v ,  u >. ]  ~R  ) )
53, 4imbi12d 233 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  f  -> 
( ( ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\  [
<. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  [ <. x ,  y
>. ]  ~R  <R  [ <. v ,  u >. ]  ~R  ) 
<->  ( ( f  <R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  f  <R  [ <. v ,  u >. ]  ~R  )
) )
6 breq2 3903 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( f  <R  [ <. z ,  w >. ]  ~R  <->  f 
<R  g ) )
7 breq1 3902 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. v ,  u >. ]  ~R  <->  g  <R  [ <. v ,  u >. ]  ~R  ) )
86, 7anbi12d 464 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( ( f  <R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  <->  ( f  <R  g  /\  g  <R  [ <. v ,  u >. ]  ~R  )
) )
98imbi1d 230 . 2  |-  ( [
<. z ,  w >. ]  ~R  =  g  -> 
( ( ( f 
<R  [ <. z ,  w >. ]  ~R  /\  [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  f  <R  [ <. v ,  u >. ]  ~R  )  <->  ( ( f  <R  g  /\  g  <R  [ <. v ,  u >. ]  ~R  )  ->  f  <R  [ <. v ,  u >. ]  ~R  ) ) )
10 breq2 3903 . . . 4  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( g  <R  [ <. v ,  u >. ]  ~R  <->  g 
<R  h ) )
1110anbi2d 459 . . 3  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( ( f  <R 
g  /\  g  <R  [
<. v ,  u >. ]  ~R  )  <->  ( f  <R  g  /\  g  <R  h ) ) )
12 breq2 3903 . . 3  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( f  <R  [ <. v ,  u >. ]  ~R  <->  f 
<R  h ) )
1311, 12imbi12d 233 . 2  |-  ( [
<. v ,  u >. ]  ~R  =  h  -> 
( ( ( f 
<R  g  /\  g  <R  [ <. v ,  u >. ]  ~R  )  -> 
f  <R  [ <. v ,  u >. ]  ~R  )  <->  ( ( f  <R  g  /\  g  <R  h )  ->  f  <R  h
) ) )
14 ltsrprg 7523 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
) )
15143adant3 986 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
) )
16 ltaprg 7395 . . . . . . . 8  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
r  <P  s  <->  ( t  +P.  r )  <P  (
t  +P.  s )
) )
1716adantl 275 . . . . . . 7  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( r  e.  P.  /\  s  e. 
P.  /\  t  e.  P. ) )  ->  (
r  <P  s  <->  ( t  +P.  r )  <P  (
t  +P.  s )
) )
18 simp1l 990 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  x  e.  P. )
19 simp2r 993 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  w  e.  P. )
20 addclpr 7313 . . . . . . . 8  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  e.  P. )
2118, 19, 20syl2anc 408 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  +P.  w )  e.  P. )
22 simp1r 991 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  y  e.  P. )
23 simp2l 992 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  z  e.  P. )
24 addclpr 7313 . . . . . . . 8  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  e.  P. )
2522, 23, 24syl2anc 408 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  +P.  z )  e.  P. )
26 simp3r 995 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  u  e.  P. )
27 addcomprg 7354 . . . . . . . 8  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  +P.  s
)  =  ( s  +P.  r ) )
2827adantl 275 . . . . . . 7  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( r  e.  P.  /\  s  e. 
P. ) )  -> 
( r  +P.  s
)  =  ( s  +P.  r ) )
2917, 21, 25, 26, 28caovord2d 5908 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  +P.  w )  <P  ( y  +P.  z
)  <->  ( ( x  +P.  w )  +P.  u )  <P  (
( y  +P.  z
)  +P.  u )
) )
30 addassprg 7355 . . . . . . . 8  |-  ( ( x  e.  P.  /\  w  e.  P.  /\  u  e.  P. )  ->  (
( x  +P.  w
)  +P.  u )  =  ( x  +P.  ( w  +P.  u ) ) )
3118, 19, 26, 30syl3anc 1201 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  +P.  w )  +P.  u )  =  ( x  +P.  ( w  +P.  u ) ) )
32 addassprg 7355 . . . . . . . 8  |-  ( ( y  e.  P.  /\  z  e.  P.  /\  u  e.  P. )  ->  (
( y  +P.  z
)  +P.  u )  =  ( y  +P.  ( z  +P.  u
) ) )
3322, 23, 26, 32syl3anc 1201 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
y  +P.  z )  +P.  u )  =  ( y  +P.  ( z  +P.  u ) ) )
3431, 33breq12d 3912 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( x  +P.  w
)  +P.  u )  <P  ( ( y  +P.  z )  +P.  u
)  <->  ( x  +P.  ( w  +P.  u ) )  <P  ( y  +P.  ( z  +P.  u
) ) ) )
3529, 34bitrd 187 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  +P.  w )  <P  ( y  +P.  z
)  <->  ( x  +P.  ( w  +P.  u ) )  <P  ( y  +P.  ( z  +P.  u
) ) ) )
3615, 35bitrd 187 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  ( w  +P.  u
) )  <P  (
y  +P.  ( z  +P.  u ) ) ) )
37 ltsrprg 7523 . . . . . 6  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( z  +P.  u )  <P  (
w  +P.  v )
) )
38373adant1 984 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( z  +P.  u )  <P  (
w  +P.  v )
) )
39 addclpr 7313 . . . . . . 7  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  +P.  u
)  e.  P. )
4023, 26, 39syl2anc 408 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( z  +P.  u )  e.  P. )
41 simp3l 994 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  v  e.  P. )
42 addclpr 7313 . . . . . . 7  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  +P.  v
)  e.  P. )
4319, 41, 42syl2anc 408 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  +P.  v )  e.  P. )
44 ltaprg 7395 . . . . . 6  |-  ( ( ( z  +P.  u
)  e.  P.  /\  ( w  +P.  v )  e.  P.  /\  y  e.  P. )  ->  (
( z  +P.  u
)  <P  ( w  +P.  v )  <->  ( y  +P.  ( z  +P.  u
) )  <P  (
y  +P.  ( w  +P.  v ) ) ) )
4540, 43, 22, 44syl3anc 1201 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
z  +P.  u )  <P  ( w  +P.  v
)  <->  ( y  +P.  ( z  +P.  u
) )  <P  (
y  +P.  ( w  +P.  v ) ) ) )
4638, 45bitrd 187 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( y  +P.  ( z  +P.  u
) )  <P  (
y  +P.  ( w  +P.  v ) ) ) )
4736, 46anbi12d 464 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\ 
[ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  ) 
<->  ( ( x  +P.  ( w  +P.  u ) )  <P  ( y  +P.  ( z  +P.  u
) )  /\  (
y  +P.  ( z  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
) ) ) )
48 ltsopr 7372 . . . . 5  |-  <P  Or  P.
49 ltrelpr 7281 . . . . 5  |-  <P  C_  ( P.  X.  P. )
5048, 49sotri 4904 . . . 4  |-  ( ( ( x  +P.  (
w  +P.  u )
)  <P  ( y  +P.  ( z  +P.  u
) )  /\  (
y  +P.  ( z  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
) )  ->  (
x  +P.  ( w  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
) )
51 addclpr 7313 . . . . . . . 8  |-  ( ( x  e.  P.  /\  u  e.  P. )  ->  ( x  +P.  u
)  e.  P. )
5218, 26, 51syl2anc 408 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  +P.  u )  e.  P. )
53 addclpr 7313 . . . . . . . 8  |-  ( ( y  e.  P.  /\  v  e.  P. )  ->  ( y  +P.  v
)  e.  P. )
5422, 41, 53syl2anc 408 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  +P.  v )  e.  P. )
55 ltaprg 7395 . . . . . . 7  |-  ( ( ( x  +P.  u
)  e.  P.  /\  ( y  +P.  v
)  e.  P.  /\  w  e.  P. )  ->  ( ( x  +P.  u )  <P  (
y  +P.  v )  <->  ( w  +P.  ( x  +P.  u ) ) 
<P  ( w  +P.  (
y  +P.  v )
) ) )
5652, 54, 19, 55syl3anc 1201 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  +P.  u )  <P  ( y  +P.  v
)  <->  ( w  +P.  ( x  +P.  u ) )  <P  ( w  +P.  ( y  +P.  v
) ) ) )
5756biimprd 157 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
w  +P.  ( x  +P.  u ) )  <P 
( w  +P.  (
y  +P.  v )
)  ->  ( x  +P.  u )  <P  (
y  +P.  v )
) )
58 addassprg 7355 . . . . . . . 8  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
( r  +P.  s
)  +P.  t )  =  ( r  +P.  ( s  +P.  t
) ) )
5958adantl 275 . . . . . . 7  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  /\  ( r  e.  P.  /\  s  e. 
P.  /\  t  e.  P. ) )  ->  (
( r  +P.  s
)  +P.  t )  =  ( r  +P.  ( s  +P.  t
) ) )
6018, 19, 26, 28, 59caov12d 5920 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( x  +P.  ( w  +P.  u
) )  =  ( w  +P.  ( x  +P.  u ) ) )
6122, 19, 41, 28, 59caov12d 5920 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( y  +P.  ( w  +P.  v
) )  =  ( w  +P.  ( y  +P.  v ) ) )
6260, 61breq12d 3912 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  +P.  ( w  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
)  <->  ( w  +P.  ( x  +P.  u ) )  <P  ( w  +P.  ( y  +P.  v
) ) ) )
63 ltsrprg 7523 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( x  +P.  u )  <P  (
y  +P.  v )
) )
64633adant2 985 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  <->  ( x  +P.  u )  <P  (
y  +P.  v )
) )
6557, 62, 643imtr4d 202 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
x  +P.  ( w  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
)  ->  [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  ) )
6650, 65syl5 32 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( (
( x  +P.  (
w  +P.  u )
)  <P  ( y  +P.  ( z  +P.  u
) )  /\  (
y  +P.  ( z  +P.  u ) )  <P 
( y  +P.  (
w  +P.  v )
) )  ->  [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  ) )
6747, 66sylbid 149 . 2  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  /\ 
[ <. z ,  w >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  )  ->  [ <. x ,  y >. ]  ~R  <R  [ <. v ,  u >. ]  ~R  ) )
681, 5, 9, 13, 673ecoptocl 6486 1  |-  ( ( f  e.  R.  /\  g  e.  R.  /\  h  e.  R. )  ->  (
( f  <R  g  /\  g  <R  h )  ->  f  <R  h
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   <.cop 3500   class class class wbr 3899  (class class class)co 5742   [cec 6395   P.cnp 7067    +P. cpp 7069    <P cltp 7071    ~R cer 7072   R.cnr 7073    <R cltr 7079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-iltp 7246  df-enr 7502  df-nr 7503  df-ltr 7506
This theorem is referenced by:  ltposr  7539
  Copyright terms: Public domain W3C validator