MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elicc4 Structured version   Unicode version

Theorem elicc4 11460
Description: Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
Assertion
Ref Expression
elicc4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A [,] B )  <->  ( A  <_  C  /\  C  <_  B ) ) )

Proof of Theorem elicc4
StepHypRef Expression
1 elicc1 11442 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
2 3anass 969 . . . 4  |-  ( ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  <->  ( C  e.  RR*  /\  ( A  <_  C  /\  C  <_  B ) ) )
31, 2syl6bb 261 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  ( A  <_  C  /\  C  <_  B ) ) ) )
43baibd 900 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* )  ->  ( C  e.  ( A [,] B )  <-> 
( A  <_  C  /\  C  <_  B ) ) )
543impa 1183 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A [,] B )  <->  ( A  <_  C  /\  C  <_  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758   class class class wbr 4387  (class class class)co 6187   RR*cxr 9515    <_ cle 9517   [,]cicc 11401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-xr 9520  df-icc 11405
This theorem is referenced by:  elicc4abs  12906  xrge0addass  26282  esumle  26639  esumlef  26644  sin2h  28557  cos2h  28558  tan2h  28559
  Copyright terms: Public domain W3C validator