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Theorem gt-lth 48158
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 ∈ V
gt-lth.2 𝐵 ∈ V
Assertion
Ref Expression
gt-lth (𝐴 > 𝐵𝐵 < 𝐴)

Proof of Theorem gt-lth
StepHypRef Expression
1 df-gt 48154 . . 3 > = <
21breqi 5154 . 2 (𝐴 > 𝐵𝐴 < 𝐵)
3 gt-lth.1 . . 3 𝐴 ∈ V
4 gt-lth.2 . . 3 𝐵 ∈ V
53, 4brcnv 5885 . 2 (𝐴 < 𝐵𝐵 < 𝐴)
62, 5bitri 275 1 (𝐴 > 𝐵𝐵 < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2099  Vcvv 3471   class class class wbr 5148  ccnv 5677   < clt 11279   > cgt 48152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-cnv 5686  df-gt 48154
This theorem is referenced by:  ex-gt  48159
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