Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gte-lte Structured version   Visualization version   GIF version

Theorem gte-lte 47064
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 47062 . . 3 ≥ =
21breqi 5109 . 2 (𝐴𝐵𝐴𝐵)
3 brcnvg 5833 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
42, 3bitrid 282 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  Vcvv 3443   class class class wbr 5103  ccnv 5630  cle 11148  cge-real 47060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-cnv 5639  df-gte 47062
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator