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Theorem gt-lt 48150
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 48148 . . 3 > = <
21breqi 5148 . 2 (𝐴 > 𝐵𝐴 < 𝐵)
3 brcnvg 5876 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 < 𝐵𝐵 < 𝐴))
42, 3bitrid 283 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 > 𝐵𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2099  Vcvv 3470   class class class wbr 5142  ccnv 5671   < clt 11272   > cgt 48146
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 5293  ax-nul 5300  ax-pr 5423
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 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-cnv 5680  df-gt 48148
This theorem is referenced by: (None)
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