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Theorem gte-lteh 49245
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 49241 . . 3 ≥ =
21breqi 5149 . 2 (𝐴𝐵𝐴𝐵)
3 gte-lteh.1 . . 3 𝐴 ∈ V
4 gte-lteh.2 . . 3 𝐵 ∈ V
53, 4brcnv 5893 . 2 (𝐴𝐵𝐵𝐴)
62, 5bitri 275 1 (𝐴𝐵𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2108  Vcvv 3480   class class class wbr 5143  ccnv 5684  cle 11296  cge-real 49239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-gte 49241
This theorem is referenced by:  ex-gte  49248
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