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Theorem aptipr 7200
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 943 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  A  e.  P. )
2 simp2 944 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  B  e.  P. )
3 ioran 704 . . . . . . 7  |-  ( -.  ( A  <P  B  \/  B  <P  A )  <->  ( -.  A  <P  B  /\  -.  B  <P  A ) )
43biimpi 118 . . . . . 6  |-  ( -.  ( A  <P  B  \/  B  <P  A )  -> 
( -.  A  <P  B  /\  -.  B  <P  A ) )
543ad2ant3 966 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( -.  A  <P  B  /\  -.  B  <P  A ) )
65simprd 112 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  -.  B  <P  A )
7 aptiprleml 7198 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 1st `  A
)  C_  ( 1st `  B ) )
81, 2, 6, 7syl3anc 1174 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  A
)  C_  ( 1st `  B ) )
95simpld 110 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  -.  A  <P  B )
10 aptiprleml 7198 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P.  /\  -.  A  <P  B )  -> 
( 1st `  B
)  C_  ( 1st `  A ) )
112, 1, 9, 10syl3anc 1174 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  B
)  C_  ( 1st `  A ) )
128, 11eqssd 3042 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  A
)  =  ( 1st `  B ) )
13 aptiprlemu 7199 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P.  /\  -.  A  <P  B )  -> 
( 2nd `  A
)  C_  ( 2nd `  B ) )
142, 1, 9, 13syl3anc 1174 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  A
)  C_  ( 2nd `  B ) )
15 aptiprlemu 7199 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 2nd `  B
)  C_  ( 2nd `  A ) )
161, 2, 6, 15syl3anc 1174 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  B
)  C_  ( 2nd `  A ) )
1714, 16eqssd 3042 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  A
)  =  ( 2nd `  B ) )
18 preqlu 7031 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <-> 
( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
19183adant3 963 . 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 890 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 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438    C_ wss 2999   class class class wbr 3845   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   P.cnp 6850    <P cltp 6854
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-enq0 6983  df-nq0 6984  df-0nq0 6985  df-plq0 6986  df-mq0 6987  df-inp 7025  df-iltp 7029
This theorem is referenced by:  aptisr  7324
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