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Theorem gt-lth 27897
Description: Relationship between  < and  > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
gt-lth.1  |-  A  e. 
_V
gt-lth.2  |-  B  e. 
_V
Assertion
Ref Expression
gt-lth  |-  ( A  >  B  <->  B  <  A )

Proof of Theorem gt-lth
StepHypRef Expression
1 df-gt 27893 . . 3  |-  >  =  `'  <
21breqi 4110 . 2  |-  ( A  >  B  <->  A `'  <  B )
3 gt-lth.1 . . 3  |-  A  e. 
_V
4 gt-lth.2 . . 3  |-  B  e. 
_V
53, 4brcnv 4946 . 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 1710   _Vcvv 2864   class class class wbr 4104   `'ccnv 4770    < clt 8957    > cgt 27891
This theorem is referenced by:  ex-gt  27898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-cnv 4779  df-gt 27893
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