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Theorem gte-lteh 42266
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 42262 . . 3 ≥ =
21breqi 4589 . 2 (𝐴𝐵𝐴𝐵)
3 gte-lteh.1 . . 3 𝐴 ∈ V
4 gte-lteh.2 . . 3 𝐵 ∈ V
53, 4brcnv 5227 . 2 (𝐴𝐵𝐵𝐴)
62, 5bitri 263 1 (𝐴𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 195  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:  ex-gte  42269
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