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Theorem brecop 6485
Description: Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
Hypotheses
Ref Expression
brecop.1  |-  .~  e.  _V
brecop.2  |-  .~  Er  ( G  X.  G
)
brecop.4  |-  H  =  ( ( G  X.  G ) /.  .~  )
brecop.5  |-  .<_  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) }
brecop.6  |-  ( ( ( ( z  e.  G  /\  w  e.  G )  /\  ( A  e.  G  /\  B  e.  G )
)  /\  ( (
v  e.  G  /\  u  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )  -> 
( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  ( ph  <->  ps )
) )
Assertion
Ref Expression
brecop  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( [ <. A ,  B >. ]  .~  .<_  [
<. C ,  D >. ]  .~  <->  ps ) )
Distinct variable groups:    x, y, z, w, v, u, A   
x, B, y, z, w, v, u    x, C, y, z, w, v, u    x, D, y, z, w, v, u   
x,  .~ , y, z, w, v, u    x, H, y    z, G, w, v, u    ph, x, y    ps, z, w, v, u
Allowed substitution hints:    ph( z, w, v, u)    ps( x, y)    G( x, y)    H( z, w, v, u)    .<_ ( x, y, z, w, v, u)

Proof of Theorem brecop
StepHypRef Expression
1 brecop.1 . . . 4  |-  .~  e.  _V
2 brecop.4 . . . 4  |-  H  =  ( ( G  X.  G ) /.  .~  )
31, 2ecopqsi 6450 . . 3  |-  ( ( A  e.  G  /\  B  e.  G )  ->  [ <. A ,  B >. ]  .~  e.  H
)
41, 2ecopqsi 6450 . . 3  |-  ( ( C  e.  G  /\  D  e.  G )  ->  [ <. C ,  D >. ]  .~  e.  H
)
5 df-br 3898 . . . . 5  |-  ( [
<. A ,  B >. ]  .~  .<_  [ <. C ,  D >. ]  .~  <->  <. [ <. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  .<_  )
6 brecop.5 . . . . . 6  |-  .<_  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) }
76eleq2i 2182 . . . . 5  |-  ( <. [ <. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  .<_  <->  <. [
<. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) } )
85, 7bitri 183 . . . 4  |-  ( [
<. A ,  B >. ]  .~  .<_  [ <. C ,  D >. ]  .~  <->  <. [ <. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) } )
9 eqeq1 2122 . . . . . . . 8  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( x  =  [ <. z ,  w >. ]  .~  <->  [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  ) )
109anbi1d 458 . . . . . . 7  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  <->  ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  ) ) )
1110anbi1d 458 . . . . . 6  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph )  <->  ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
12114exbidv 1824 . . . . 5  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph )  <->  E. z E. w E. v E. u ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
13 eqeq1 2122 . . . . . . . 8  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( y  =  [ <. v ,  u >. ]  .~  <->  [
<. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  ) )
1413anbi2d 457 . . . . . . 7  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  <->  ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  ) ) )
1514anbi1d 458 . . . . . 6  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph )  <->  ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
16154exbidv 1824 . . . . 5  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( E. z E. w E. v E. u ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph )  <->  E. z E. w E. v E. u ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
1712, 16opelopab2 4160 . . . 4  |-  ( ( [ <. A ,  B >. ]  .~  e.  H  /\  [ <. C ,  D >. ]  .~  e.  H
)  ->  ( <. [
<. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) }  <->  E. z E. w E. v E. u ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
188, 17syl5bb 191 . . 3  |-  ( ( [ <. A ,  B >. ]  .~  e.  H  /\  [ <. C ,  D >. ]  .~  e.  H
)  ->  ( [ <. A ,  B >. ]  .~  .<_  [ <. C ,  D >. ]  .~  <->  E. z E. w E. v E. u ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
193, 4, 18syl2an 285 . 2  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( [ <. A ,  B >. ]  .~  .<_  [
<. C ,  D >. ]  .~  <->  E. z E. w E. v E. u ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
20 opeq12 3675 . . . . . 6  |-  ( ( z  =  A  /\  w  =  B )  -> 
<. z ,  w >.  = 
<. A ,  B >. )
2120eceq1d 6431 . . . . 5  |-  ( ( z  =  A  /\  w  =  B )  ->  [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  )
22 opeq12 3675 . . . . . 6  |-  ( ( v  =  C  /\  u  =  D )  -> 
<. v ,  u >.  = 
<. C ,  D >. )
2322eceq1d 6431 . . . . 5  |-  ( ( v  =  C  /\  u  =  D )  ->  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )
2421, 23anim12i 334 . . . 4  |-  ( ( ( z  =  A  /\  w  =  B )  /\  ( v  =  C  /\  u  =  D ) )  -> 
( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ) )
25 opelxpi 4539 . . . . . . . 8  |-  ( ( A  e.  G  /\  B  e.  G )  -> 
<. A ,  B >.  e.  ( G  X.  G
) )
26 opelxp 4537 . . . . . . . . 9  |-  ( <.
z ,  w >.  e.  ( G  X.  G
)  <->  ( z  e.  G  /\  w  e.  G ) )
27 brecop.2 . . . . . . . . . . 11  |-  .~  Er  ( G  X.  G
)
2827a1i 9 . . . . . . . . . 10  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  .~  Er  ( G  X.  G ) )
29 id 19 . . . . . . . . . 10  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  )
3028, 29ereldm 6438 . . . . . . . . 9  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  ( <. z ,  w >.  e.  ( G  X.  G )  <->  <. A ,  B >.  e.  ( G  X.  G ) ) )
3126, 30syl5bbr 193 . . . . . . . 8  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  ( ( z  e.  G  /\  w  e.  G )  <->  <. A ,  B >.  e.  ( G  X.  G ) ) )
3225, 31syl5ibr 155 . . . . . . 7  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  ( ( A  e.  G  /\  B  e.  G )  ->  (
z  e.  G  /\  w  e.  G )
) )
33 opelxpi 4539 . . . . . . . 8  |-  ( ( C  e.  G  /\  D  e.  G )  -> 
<. C ,  D >.  e.  ( G  X.  G
) )
34 opelxp 4537 . . . . . . . . 9  |-  ( <.
v ,  u >.  e.  ( G  X.  G
)  <->  ( v  e.  G  /\  u  e.  G ) )
3527a1i 9 . . . . . . . . . 10  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  .~  Er  ( G  X.  G ) )
36 id 19 . . . . . . . . . 10  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )
3735, 36ereldm 6438 . . . . . . . . 9  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  ( <. v ,  u >.  e.  ( G  X.  G )  <->  <. C ,  D >.  e.  ( G  X.  G ) ) )
3834, 37syl5bbr 193 . . . . . . . 8  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  ( ( v  e.  G  /\  u  e.  G )  <->  <. C ,  D >.  e.  ( G  X.  G ) ) )
3933, 38syl5ibr 155 . . . . . . 7  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  ( ( C  e.  G  /\  D  e.  G )  ->  (
v  e.  G  /\  u  e.  G )
) )
4032, 39im2anan9 570 . . . . . 6  |-  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ( (
z  e.  G  /\  w  e.  G )  /\  ( v  e.  G  /\  u  e.  G
) ) ) )
41 brecop.6 . . . . . . . . 9  |-  ( ( ( ( z  e.  G  /\  w  e.  G )  /\  ( A  e.  G  /\  B  e.  G )
)  /\  ( (
v  e.  G  /\  u  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )  -> 
( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  ( ph  <->  ps )
) )
4241an4s 560 . . . . . . . 8  |-  ( ( ( ( z  e.  G  /\  w  e.  G )  /\  (
v  e.  G  /\  u  e.  G )
)  /\  ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )  -> 
( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  ( ph  <->  ps )
) )
4342ex 114 . . . . . . 7  |-  ( ( ( z  e.  G  /\  w  e.  G
)  /\  ( v  e.  G  /\  u  e.  G ) )  -> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ph  <->  ps ) ) ) )
4443com13 80 . . . . . 6  |-  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ( (
( z  e.  G  /\  w  e.  G
)  /\  ( v  e.  G  /\  u  e.  G ) )  -> 
( ph  <->  ps ) ) ) )
4540, 44mpdd 41 . . . . 5  |-  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ( ph  <->  ps ) ) )
4645pm5.74d 181 . . . 4  |-  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) )  ->  ph )  <->  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ps )
) )
4724, 46cgsex4g 2695 . . 3  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( E. z E. w E. v E. u ( ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  (
( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ph ) )  <-> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ps )
) )
48 eqcom 2117 . . . . . . 7  |-  ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  <->  [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  )
49 eqcom 2117 . . . . . . 7  |-  ( [
<. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  <->  [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )
5048, 49anbi12i 453 . . . . . 6  |-  ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  <->  ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ) )
5150a1i 9 . . . . 5  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  ) 
<->  ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ) ) )
52 biimt 240 . . . . 5  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( ph  <->  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) )  ->  ph )
) )
5351, 52anbi12d 462 . . . 4  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) 
<->  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) )  ->  ph )
) ) )
54534exbidv 1824 . . 3  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( E. z E. w E. v E. u ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) 
<->  E. z E. w E. v E. u ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) )  ->  ph )
) ) )
55 biimt 240 . . 3  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( ps  <->  ( (
( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  ->  ps ) ) )
5647, 54, 553bitr4d 219 . 2  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( E. z E. w E. v E. u ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) 
<->  ps ) )
5719, 56bitrd 187 1  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( [ <. A ,  B >. ]  .~  .<_  [
<. C ,  D >. ]  .~  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   _Vcvv 2658   <.cop 3498   class class class wbr 3897   {copab 3956    X. cxp 4505    Er wer 6392   [cec 6393   /.cqs 6394
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-er 6395  df-ec 6397  df-qs 6401
This theorem is referenced by:  ordpipqqs  7146  ltsrprg  7519
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