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Theorem gte-lte 42264
 Description: Simple relationship between ≤ and ≥. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
Assertion
Ref Expression
gte-lte ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))

Proof of Theorem gte-lte
StepHypRef Expression
1 df-gte 42262 . . 3 ≥ =
21breqi 4589 . 2 (𝐴𝐵𝐴𝐵)
3 brcnvg 5225 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
42, 3syl5bb 271 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  ◡ccnv 5037   ≤ cle 9954   ≥ cge-real 42260 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-gte 42262 This theorem is referenced by: (None)
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