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Theorem gte-lte 42236
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 42234 . . 3 ≥ =
21breqi 4657 . 2 (𝐴𝐵𝐴𝐵)
3 brcnvg 5301 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
42, 3syl5bb 272 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1989  Vcvv 3198   class class class wbr 4651  ccnv 5111  cle 10072  cge-real 42232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-cnv 5120  df-gte 42234
This theorem is referenced by: (None)
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