Theorem gte-lte 49706

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 49704	. . 3 ⊢ ≥ = ◡ ≤
2	1	breqi 5108	. 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵)
3		brcnvg 5833	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴))
4	2, 3	bitrid 283	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3444   class class class wbr 5102  ^◡ccnv 5630   ≤ cle 11185   ≥ cge-real 49702

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-cnv 5639  df-gte 49704

This theorem is referenced by: (None)