MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ubioc1 Structured version   Unicode version

Theorem ubioc1 10996
Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 11045. (Contributed by FL, 29-May-2014.)
Assertion
Ref Expression
ubioc1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  e.  ( A (,] B
) )

Proof of Theorem ubioc1
StepHypRef Expression
1 simp2 959 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  e.  RR* )
2 simp3 960 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <  B )
3 xrleid 10774 . . 3  |-  ( B  e.  RR*  ->  B  <_  B )
433ad2ant2 980 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  <_  B )
5 elioc1 10989 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  e.  ( A (,] B )  <->  ( B  e.  RR*  /\  A  < 
B  /\  B  <_  B ) ) )
653adant3 978 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( B  e.  ( A (,] B )  <->  ( B  e.  RR*  /\  A  < 
B  /\  B  <_  B ) ) )
71, 2, 4, 6mpbir3and 1138 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  e.  ( A (,] B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    e. wcel 1727   class class class wbr 4237  (class class class)co 6110   RR*cxr 9150    < clt 9151    <_ cle 9152   (,]cioc 10948
This theorem is referenced by:  xrlimcnp  20838  pnfneige0  24367  lmxrge0  24368  dvreasin  26328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-pre-lttri 9095  ax-pre-lttrn 9096
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-ioc 10952
  Copyright terms: Public domain W3C validator