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Theorem aptisr 7017
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 6966 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 3796 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
3 breq2 3797 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  A ) )
42, 3orbi12d 740 . . . . 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 625 . . . 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 2088 . . . 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 3797 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
9 breq1 3796 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  <R  A  <-> 
B  <R  A ) )
108, 9orbi12d 740 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  <R  [
<. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  ( A  <R  B  \/  B  <R  A ) ) )
1110notbid 625 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( -.  ( A 
<R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  -.  ( A  <R  B  \/  B  <R  A ) ) )
12 eqeq2 2091 . . . 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 6830 . . . . . . . . 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 6830 . . . . . . . . 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 3806 . . . . . . 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 737 . . . . . 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 625 . . . . 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 6789 . . . . . . 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 6789 . . . . . . 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 6893 . . . . . . 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 1141 . . . . . 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 6986 . . . . . 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 6986 . . . . . . 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 740 . . . . 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 625 . . . 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 6975 . . . 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 6260 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( -.  ( A 
<R  B  \/  B  <R  A )  ->  A  =  B ) )
37363impia 1136 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 662    /\ w3a 920    = wceq 1285    e. wcel 1434   <.cop 3409   class class class wbr 3793  (class class class)co 5543   [cec 6170   P.cnp 6543    +P. cpp 6545    <P cltp 6547    ~R cer 6548   R.cnr 6549    <R cltr 6555
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722  df-enr 6965  df-nr 6966  df-ltr 6969
This theorem is referenced by:  axpre-apti  7113
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