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Theorem gt-lth 28197
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 28193 . . 3  |-  >  =  `'  <
21breqi 4029 . 2  |-  ( A  >  B  <->  A `'  <  B )
3 gt-lth.1 . . 3  |-  A  e. 
_V
4 gt-lth.2 . . 3  |-  B  e. 
_V
53, 4brcnv 4864 . 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 1684   _Vcvv 2788   class class class wbr 4023   `'ccnv 4688    < clt 8867    > cgt 28191
This theorem is referenced by:  ex-gt  28198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-gt 28193
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