ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  aptipr Unicode version

Theorem aptipr 7261
Description: Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)
Assertion
Ref Expression
aptipr  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  A  =  B )

Proof of Theorem aptipr
StepHypRef Expression
1 simp1 944 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  A  e.  P. )
2 simp2 945 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  B  e.  P. )
3 ioran 705 . . . . . . 7  |-  ( -.  ( A  <P  B  \/  B  <P  A )  <->  ( -.  A  <P  B  /\  -.  B  <P  A ) )
43biimpi 119 . . . . . 6  |-  ( -.  ( A  <P  B  \/  B  <P  A )  -> 
( -.  A  <P  B  /\  -.  B  <P  A ) )
543ad2ant3 967 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( -.  A  <P  B  /\  -.  B  <P  A ) )
65simprd 113 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  -.  B  <P  A )
7 aptiprleml 7259 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 1st `  A
)  C_  ( 1st `  B ) )
81, 2, 6, 7syl3anc 1175 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  A
)  C_  ( 1st `  B ) )
95simpld 111 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  -.  A  <P  B )
10 aptiprleml 7259 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P.  /\  -.  A  <P  B )  -> 
( 1st `  B
)  C_  ( 1st `  A ) )
112, 1, 9, 10syl3anc 1175 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  B
)  C_  ( 1st `  A ) )
128, 11eqssd 3043 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  A
)  =  ( 1st `  B ) )
13 aptiprlemu 7260 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P.  /\  -.  A  <P  B )  -> 
( 2nd `  A
)  C_  ( 2nd `  B ) )
142, 1, 9, 13syl3anc 1175 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  A
)  C_  ( 2nd `  B ) )
15 aptiprlemu 7260 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 2nd `  B
)  C_  ( 2nd `  A ) )
161, 2, 6, 15syl3anc 1175 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  B
)  C_  ( 2nd `  A ) )
1714, 16eqssd 3043 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  A
)  =  ( 2nd `  B ) )
18 preqlu 7092 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <-> 
( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
19183adant3 964 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( A  =  B  <->  ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A
)  =  ( 2nd `  B ) ) ) )
2012, 17, 19mpbir2and 891 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 665    /\ w3a 925    = wceq 1290    e. wcel 1439    C_ wss 3000   class class class wbr 3851   ` cfv 5028   1stc1st 5923   2ndc2nd 5924   P.cnp 6911    <P cltp 6915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-2o 6196  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-enq0 7044  df-nq0 7045  df-0nq0 7046  df-plq0 7047  df-mq0 7048  df-inp 7086  df-iltp 7090
This theorem is referenced by:  aptisr  7385
  Copyright terms: Public domain W3C validator