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Theorem gt-lth 28470
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 28466 . . 3  |-  >  =  `'  <
21breqi 4218 . 2  |-  ( A  >  B  <->  A `'  <  B )
3 gt-lth.1 . . 3  |-  A  e. 
_V
4 gt-lth.2 . . 3  |-  B  e. 
_V
53, 4brcnv 5055 . 2  |-  ( A `'  <  B  <->  B  <  A )
62, 5bitri 241 1  |-  ( A  >  B  <->  B  <  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    e. wcel 1725   _Vcvv 2956   class class class wbr 4212   `'ccnv 4877    < clt 9120    > cgt 28464
This theorem is referenced by:  ex-gt  28471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-cnv 4886  df-gt 28466
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