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Theorem gt-lt 41789
 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 41787 . . 3 > = <
21breqi 4629 . 2 (𝐴 > 𝐵𝐴 < 𝐵)
3 brcnvg 5273 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 < 𝐵𝐵 < 𝐴))
42, 3syl5bb 272 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵𝐵 < 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1987  Vcvv 3190   class class class wbr 4623  ◡ccnv 5083   < clt 10034   > cgt 41785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-cnv 5092  df-gt 41787 This theorem is referenced by: (None)
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