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Theorem gt-lt 44752
Description: Simple relationship between < and >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
gt-lt ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵𝐵 < 𝐴))

Proof of Theorem gt-lt
StepHypRef Expression
1 df-gt 44750 . . 3 > = <
21breqi 5063 . 2 (𝐴 > 𝐵𝐴 < 𝐵)
3 brcnvg 5743 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 < 𝐵𝐵 < 𝐴))
42, 3syl5bb 284 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  Vcvv 3492   class class class wbr 5057  ccnv 5547   < clt 10663   > cgt 44748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-cnv 5556  df-gt 44750
This theorem is referenced by: (None)
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