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

Theorem ubioc1 10721
Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 10769. (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 956 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  e.  RR* )
2 simp3 957 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  A  <  B )
3 xrleid 10500 . . 3  |-  ( B  e.  RR*  ->  B  <_  B )
433ad2ant2 977 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  <_  B )
5 elioc1 10714 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  e.  ( A (,] B )  <->  ( B  e.  RR*  /\  A  < 
B  /\  B  <_  B ) ) )
653adant3 975 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  ( B  e.  ( A (,] B )  <->  ( B  e.  RR*  /\  A  < 
B  /\  B  <_  B ) ) )
71, 2, 4, 6mpbir3and 1135 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  < 
B )  ->  B  e.  ( A (,] B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RR*cxr 8882    < clt 8883    <_ cle 8884   (,]cioc 10673
This theorem is referenced by:  xrlimcnp  20279  pnfneige0  23389  lmxrge0  23390  dvreasin  25026  intvconlem1  25806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-ioc 10677
  Copyright terms: Public domain W3C validator