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Theorem aptipr 7590
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 992 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  A  e.  P. )
2 simp2 993 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  B  e.  P. )
3 ioran 747 . . . . . . 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 1015 . . . . 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 7588 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 1st `  A
)  C_  ( 1st `  B ) )
81, 2, 6, 7syl3anc 1233 . . 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 7588 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P.  /\  -.  A  <P  B )  -> 
( 1st `  B
)  C_  ( 1st `  A ) )
112, 1, 9, 10syl3anc 1233 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  B
)  C_  ( 1st `  A ) )
128, 11eqssd 3164 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 1st `  A
)  =  ( 1st `  B ) )
13 aptiprlemu 7589 . . . 4  |-  ( ( B  e.  P.  /\  A  e.  P.  /\  -.  A  <P  B )  -> 
( 2nd `  A
)  C_  ( 2nd `  B ) )
142, 1, 9, 13syl3anc 1233 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  A
)  C_  ( 2nd `  B ) )
15 aptiprlemu 7589 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  B  <P  A )  -> 
( 2nd `  B
)  C_  ( 2nd `  A ) )
161, 2, 6, 15syl3anc 1233 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  B
)  C_  ( 2nd `  A ) )
1714, 16eqssd 3164 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  -.  ( A  <P  B  \/  B  <P  A ) )  ->  ( 2nd `  A
)  =  ( 2nd `  B ) )
18 preqlu 7421 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  =  B  <-> 
( ( 1st `  A
)  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B
) ) ) )
19183adant3 1012 . 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 939 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 703    /\ w3a 973    = wceq 1348    e. wcel 2141    C_ wss 3121   class class class wbr 3987   ` cfv 5196   1stc1st 6114   2ndc2nd 6115   P.cnp 7240    <P cltp 7244
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-2o 6393  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-enq0 7373  df-nq0 7374  df-0nq0 7375  df-plq0 7376  df-mq0 7377  df-inp 7415  df-iltp 7419
This theorem is referenced by:  aptisr  7728
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