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Theorem gt-lth 28184
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 28180 . . 3  |-  >  =  `'  <
21breqi 4178 . 2  |-  ( A  >  B  <->  A `'  <  B )
3 gt-lth.1 . . 3  |-  A  e. 
_V
4 gt-lth.2 . . 3  |-  B  e. 
_V
53, 4brcnv 5014 . 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 1721   _Vcvv 2916   class class class wbr 4172   `'ccnv 4836    < clt 9076    > cgt 28178
This theorem is referenced by:  ex-gt  28185
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-cnv 4845  df-gt 28180
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