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Theorem aptisr 7303
Description: Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
aptisr  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  -.  ( A  <R  B  \/  B  <R  A ) )  ->  A  =  B )

Proof of Theorem aptisr
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7252 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 3840 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
3 breq2 3841 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  A ) )
42, 3orbi12d 742 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) 
<->  ( A  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A ) ) )
54notbid 627 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( -.  ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [
<. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) 
<->  -.  ( A  <R  [
<. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A ) ) )
6 eqeq1 2094 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  <->  A  =  [ <. z ,  w >. ]  ~R  ) )
75, 6imbi12d 232 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( -.  ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  )  ->  [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  )  <->  ( -.  ( A  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  ->  A  =  [ <. z ,  w >. ]  ~R  ) ) )
8 breq2 3841 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
9 breq1 3840 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  <R  A  <-> 
B  <R  A ) )
108, 9orbi12d 742 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  <R  [
<. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  ( A  <R  B  \/  B  <R  A ) ) )
1110notbid 627 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( -.  ( A 
<R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  -.  ( A  <R  B  \/  B  <R  A ) ) )
12 eqeq2 2097 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  =  [ <. z ,  w >. ]  ~R  <->  A  =  B
) )
1311, 12imbi12d 232 . . 3  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( -.  ( A  <R  [ <. z ,  w >. ]  ~R  \/  [
<. z ,  w >. ]  ~R  <R  A )  ->  A  =  [ <. z ,  w >. ]  ~R  ) 
<->  ( -.  ( A 
<R  B  \/  B  <R  A )  ->  A  =  B ) ) )
14 addcomprg 7116 . . . . . . . . 9  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  =  ( z  +P.  y ) )
1514ad2ant2lr 494 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  +P.  z )  =  ( z  +P.  y ) )
16 addcomprg 7116 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  =  ( w  +P.  x ) )
1716ad2ant2rl 495 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  +P.  w )  =  ( w  +P.  x ) )
1815, 17breq12d 3850 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
y  +P.  z )  <P  ( x  +P.  w
)  <->  ( z  +P.  y )  <P  (
w  +P.  x )
) )
1918orbi2d 739 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( (
( x  +P.  w
)  <P  ( y  +P.  z )  \/  (
y  +P.  z )  <P  ( x  +P.  w
) )  <->  ( (
x  +P.  w )  <P  ( y  +P.  z
)  \/  ( z  +P.  y )  <P 
( w  +P.  x
) ) ) )
2019notbid 627 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( -.  ( ( x  +P.  w )  <P  (
y  +P.  z )  \/  ( y  +P.  z
)  <P  ( x  +P.  w ) )  <->  -.  (
( x  +P.  w
)  <P  ( y  +P.  z )  \/  (
z  +P.  y )  <P  ( w  +P.  x
) ) ) )
21 addclpr 7075 . . . . . . 7  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  e.  P. )
2221ad2ant2rl 495 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  +P.  w )  e.  P. )
23 addclpr 7075 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  e.  P. )
2423ad2ant2lr 494 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  +P.  z )  e.  P. )
25 aptipr 7179 . . . . . . 7  |-  ( ( ( x  +P.  w
)  e.  P.  /\  ( y  +P.  z
)  e.  P.  /\  -.  ( ( x  +P.  w )  <P  (
y  +P.  z )  \/  ( y  +P.  z
)  <P  ( x  +P.  w ) ) )  ->  ( x  +P.  w )  =  ( y  +P.  z ) )
26253expia 1145 . . . . . 6  |-  ( ( ( x  +P.  w
)  e.  P.  /\  ( y  +P.  z
)  e.  P. )  ->  ( -.  ( ( x  +P.  w ) 
<P  ( y  +P.  z
)  \/  ( y  +P.  z )  <P 
( x  +P.  w
) )  ->  (
x  +P.  w )  =  ( y  +P.  z ) ) )
2722, 24, 26syl2anc 403 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( -.  ( ( x  +P.  w )  <P  (
y  +P.  z )  \/  ( y  +P.  z
)  <P  ( x  +P.  w ) )  -> 
( x  +P.  w
)  =  ( y  +P.  z ) ) )
2820, 27sylbird 168 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( -.  ( ( x  +P.  w )  <P  (
y  +P.  z )  \/  ( z  +P.  y
)  <P  ( w  +P.  x ) )  -> 
( x  +P.  w
)  =  ( y  +P.  z ) ) )
29 ltsrprg 7272 . . . . . 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 )
) )
30 ltsrprg 7272 . . . . . . 7  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( z  +P.  y ) 
<P  ( w  +P.  x
) ) )
3130ancoms 264 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  <->  ( z  +P.  y ) 
<P  ( w  +P.  x
) ) )
3229, 31orbi12d 742 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) 
<->  ( ( x  +P.  w )  <P  (
y  +P.  z )  \/  ( z  +P.  y
)  <P  ( w  +P.  x ) ) ) )
3332notbid 627 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( -.  ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  ) 
<->  -.  ( ( x  +P.  w )  <P 
( y  +P.  z
)  \/  ( z  +P.  y )  <P 
( w  +P.  x
) ) ) )
34 enreceq 7261 . . . 4  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  =  ( y  +P.  z ) ) )
3528, 33, 343imtr4d 201 . . 3  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( -.  ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  [ <. x ,  y >. ]  ~R  )  ->  [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  ) )
361, 7, 13, 352ecoptocl 6360 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( -.  ( A 
<R  B  \/  B  <R  A )  ->  A  =  B ) )
37363impia 1140 1  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  -.  ( A  <R  B  \/  B  <R  A ) )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   <.cop 3444   class class class wbr 3837  (class class class)co 5634   [cec 6270   P.cnp 6829    +P. cpp 6831    <P cltp 6833    ~R cer 6834   R.cnr 6835    <R cltr 6841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008  df-enr 7251  df-nr 7252  df-ltr 7255
This theorem is referenced by:  axpre-apti  7399
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