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Theorem gt-lth 46100
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 46096 . . 3 > = <
21breqi 5059 . 2 (𝐴 > 𝐵𝐴 < 𝐵)
3 gt-lth.1 . . 3 𝐴 ∈ V
4 gt-lth.2 . . 3 𝐵 ∈ V
53, 4brcnv 5751 . 2 (𝐴 < 𝐵𝐵 < 𝐴)
62, 5bitri 278 1 (𝐴 > 𝐵𝐵 < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2110  Vcvv 3408   class class class wbr 5053  ccnv 5550   < clt 10867   > cgt 46094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-cnv 5559  df-gt 46096
This theorem is referenced by:  ex-gt  46101
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