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Theorem gt-lth 47258
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 47254 . . 3 > = <
21breqi 5112 . 2 (𝐴 > 𝐵𝐴 < 𝐵)
3 gt-lth.1 . . 3 𝐴 ∈ V
4 gt-lth.2 . . 3 𝐵 ∈ V
53, 4brcnv 5839 . 2 (𝐴 < 𝐵𝐵 < 𝐴)
62, 5bitri 275 1 (𝐴 > 𝐵𝐵 < 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2107  Vcvv 3444   class class class wbr 5106  ccnv 5633   < clt 11194   > cgt 47252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-gt 47254
This theorem is referenced by:  ex-gt  47259
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