Theorem gte-lte 50637

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 50635	. . 3 ⊢ ≥ = ◡ ≤
2	1	breqi 5113	. 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵)
3		brcnvg 5863	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴))
4	2, 3	bitrid 286	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  Vcvv 3453   class class class wbr 5107  ^◡ccnv 5658   ≤ cle 11271   ≥ cge-real 50633

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-cnv 5667  df-gte 50635

This theorem is used by: (None)