Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gte-lteh Unicode version

Theorem gte-lteh 28450
Description: Relationship between  <_ and  >_ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
gte-lteh.1  |-  A  e. 
_V
gte-lteh.2  |-  B  e. 
_V
Assertion
Ref Expression
gte-lteh  |-  ( A  >_  B  <->  B  <_  A )

Proof of Theorem gte-lteh
StepHypRef Expression
1 df-gte 28446 . . 3  |-  >_  =  `'  <_
21breqi 4045 . 2  |-  ( A  >_  B  <->  A `'  <_  B )
3 gte-lteh.1 . . 3  |-  A  e. 
_V
4 gte-lteh.2 . . 3  |-  B  e. 
_V
53, 4brcnv 4880 . 2  |-  ( A `'  <_  B  <->  B  <_  A )
62, 5bitri 240 1  |-  ( A  >_  B  <->  B  <_  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1696   _Vcvv 2801   class class class wbr 4039   `'ccnv 4704    <_ cle 8884    >_ cge-real 28444
This theorem is referenced by:  ex-gte  28453
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-gte 28446
  Copyright terms: Public domain W3C validator