Theorem gte-lte 46312

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) → (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴))

Proof of Theorem gte-lte

Step	Hyp	Ref	Expression
1		df-gte 46310	. . 3 ⊢ ≥ = ◡ ≤
2	1	breqi 5076	. 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵)
3		brcnvg 5777	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴))
4	2, 3	syl5bb 282	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070  ^◡ccnv 5579   ≤ cle 10941   ≥ cge-real 46308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-cnv 5588  df-gte 46310

This theorem is referenced by: (None)