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Theorem aptisr 7555
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 7503 . . 3  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 breq1 3902 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
3 breq2 3903 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. z ,  w >. ]  ~R  <R  [
<. x ,  y >. ]  ~R  <->  [ <. z ,  w >. ]  ~R  <R  A ) )
42, 3orbi12d 767 . . . . 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 641 . . . 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 2124 . . . 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 3903 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
9 breq1 3902 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( [ <. z ,  w >. ]  ~R  <R  A  <-> 
B  <R  A ) )
108, 9orbi12d 767 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  <R  [
<. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  ( A  <R  B  \/  B  <R  A ) ) )
1110notbid 641 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( -.  ( A 
<R  [ <. z ,  w >. ]  ~R  \/  [ <. z ,  w >. ]  ~R  <R  A )  <->  -.  ( A  <R  B  \/  B  <R  A ) ) )
12 eqeq2 2127 . . . 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 7354 . . . . . . . . 9  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  =  ( z  +P.  y ) )
1514ad2ant2lr 501 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  +P.  z )  =  ( z  +P.  y ) )
16 addcomprg 7354 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  =  ( w  +P.  x ) )
1716ad2ant2rl 502 . . . . . . . 8  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  +P.  w )  =  ( w  +P.  x ) )
1815, 17breq12d 3912 . . . . . . 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 764 . . . . . 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 641 . . . . 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 7313 . . . . . . 7  |-  ( ( x  e.  P.  /\  w  e.  P. )  ->  ( x  +P.  w
)  e.  P. )
2221ad2ant2rl 502 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( x  +P.  w )  e.  P. )
23 addclpr 7313 . . . . . . 7  |-  ( ( y  e.  P.  /\  z  e.  P. )  ->  ( y  +P.  z
)  e.  P. )
2423ad2ant2lr 501 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( y  +P.  z )  e.  P. )
25 aptipr 7417 . . . . . . 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 1168 . . . . . 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 408 . . . . 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 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 )
) )
30 ltsrprg 7523 . . . . . . 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 767 . . . . 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 641 . . . 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 7512 . . . 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 6485 . 2  |-  ( ( A  e.  R.  /\  B  e.  R. )  ->  ( -.  ( A 
<R  B  \/  B  <R  A )  ->  A  =  B ) )
37363impia 1163 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 682    /\ 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:  axpre-apti  7661
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