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Theorem gte-lteh 44226
 Description: Relationship between ≤ and ≥ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
gte-lteh.1 𝐴 ∈ V
gte-lteh.2 𝐵 ∈ V
Assertion
Ref Expression
gte-lteh (𝐴𝐵𝐵𝐴)

Proof of Theorem gte-lteh
StepHypRef Expression
1 df-gte 44222 . . 3 ≥ =
21breqi 4931 . 2 (𝐴𝐵𝐴𝐵)
3 gte-lteh.1 . . 3 𝐴 ∈ V
4 gte-lteh.2 . . 3 𝐵 ∈ V
53, 4brcnv 5599 . 2 (𝐴𝐵𝐵𝐴)
62, 5bitri 267 1 (𝐴𝐵𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∈ wcel 2051  Vcvv 3408   class class class wbr 4925  ◡ccnv 5402   ≤ cle 10473   ≥ cge-real 44220 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-cnv 5411  df-gte 44222 This theorem is referenced by:  ex-gte  44229
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