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Theorem aptisr 7610
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 7558 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 3939 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
3 breq2 3940 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  A ) )
42, 3orbi12d 783 . . . . 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 657 . . . 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 2147 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  =  [ <. z ,  w >. ]  ~R  <->  A  =  [ <. z ,  w >. ]  ~R  ) )
75, 6imbi12d 233 . . 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 3940 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
9 breq1 3939 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  <R  A  <-> 
B  <R  A ) )
108, 9orbi12d 783 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  <R  [
<. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  ( A  <R  B  \/  B  <R  A ) ) )
1110notbid 657 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( -.  ( A 
<R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  -.  ( A  <R  B  \/  B  <R  A ) ) )
12 eqeq2 2150 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  =  [ <. z ,  w >. ]  ~R  <->  A  =  B
) )
1311, 12imbi12d 233 . . 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 7409 . . . . . . . . 9  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  =  ( z  +P.  y ) )
1514ad2ant2lr 502 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  +P.  z )  =  ( z  +P.  y ) )
16 addcomprg 7409 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  =  ( w  +P.  x ) )
1716ad2ant2rl 503 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  +P.  w )  =  ( w  +P.  x ) )
1815, 17breq12d 3949 . . . . . . 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 780 . . . . . 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 657 . . . . 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 7368 . . . . . . 7  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  e.  P. )
2221ad2ant2rl 503 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  +P.  w )  e.  P. )
23 addclpr 7368 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  e.  P. )
2423ad2ant2lr 502 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  +P.  z )  e.  P. )
25 aptipr 7472 . . . . . . 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 1184 . . . . . 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 409 . . . . 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 169 . . . 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 7578 . . . . . 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 7578 . . . . . . 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 266 . . . . . 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 783 . . . . 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 657 . . . 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 7567 . . . 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 202 . . 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 6524 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( -.  ( A 
<R  B  \/  B  <R  A )  ->  A  =  B ) )
37363impia 1179 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 103    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1332    e. wcel 1481   <.cop 3534   class class class wbr 3936  (class class class)co 5781   [cec 6434   P.cnp 7122    +P. cpp 7124    <P cltp 7126    ~R cer 7127   R.cnr 7128    <R cltr 7134
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-2o 6321  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-enq0 7255  df-nq0 7256  df-0nq0 7257  df-plq0 7258  df-mq0 7259  df-inp 7297  df-iplp 7299  df-iltp 7301  df-enr 7557  df-nr 7558  df-ltr 7561
This theorem is referenced by:  axpre-apti  7716
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