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Theorem aptipr 7456
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 981 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  A  e.  P. )
2 simp2 982 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  B  e.  P. )
3 ioran 741 . . . . . . 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 1004 . . . . 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 7454 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 1st `  A
)  C_  ( 1st `  B ) )
81, 2, 6, 7syl3anc 1216 . . 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 7454 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P.  /\  -.  A  <P  B )  -> 
( 1st `  B
)  C_  ( 1st `  A ) )
112, 1, 9, 10syl3anc 1216 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  B
)  C_  ( 1st `  A ) )
128, 11eqssd 3114 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  A
)  =  ( 1st `  B ) )
13 aptiprlemu 7455 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P.  /\  -.  A  <P  B )  -> 
( 2nd `  A
)  C_  ( 2nd `  B ) )
142, 1, 9, 13syl3anc 1216 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  A
)  C_  ( 2nd `  B ) )
15 aptiprlemu 7455 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 2nd `  B
)  C_  ( 2nd `  A ) )
161, 2, 6, 15syl3anc 1216 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  B
)  C_  ( 2nd `  A ) )
1714, 16eqssd 3114 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  A
)  =  ( 2nd `  B ) )
18 preqlu 7287 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <-> 
( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
19183adant3 1001 . 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 928 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 697    /\ w3a 962    = wceq 1331    e. wcel 1480    C_ wss 3071   class class class wbr 3929   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   P.cnp 7106    <P cltp 7110
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iltp 7285
This theorem is referenced by:  aptisr  7594
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