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Theorem gte-lte 48206
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 48204 . . 3 ≥ =
21breqi 5156 . 2 (𝐴𝐵𝐴𝐵)
3 brcnvg 5884 . 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 394  wcel 2098  Vcvv 3471   class class class wbr 5150  ccnv 5679  cle 11285  cge-real 48202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-cnv 5688  df-gte 48204
This theorem is referenced by: (None)
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