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Theorem aptipr 7603
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 7601 . . . 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 7601 . . . 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 7602 . . . 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 7602 . . . 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 7434 . . 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 3989   ` cfv 5198   1stc1st 6117   2ndc2nd 6118   P.cnp 7253    <P cltp 7257
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 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iltp 7432
This theorem is referenced by:  aptisr  7741
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