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Theorem gt-lt 28405
Description: Simple relationship between  < and  >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
gt-lt  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  >  B  <->  B  <  A ) )

Proof of Theorem gt-lt
StepHypRef Expression
1 df-gt 28403 . . 3  |-  >  =  `'  <
21breqi 4210 . 2  |-  ( A  >  B  <->  A `'  <  B )
3 brcnvg 5045 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A `'  <  B  <-> 
B  <  A )
)
42, 3syl5bb 249 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  >  B  <->  B  <  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   _Vcvv 2948   class class class wbr 4204   `'ccnv 4869    < clt 9112    > cgt 28401
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-cnv 4878  df-gt 28403
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