Theorem gte-lte 44843

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 44841	. . 3 ⊢ ≥ = ◡ ≤
2	1	breqi 5072	. 2 ⊢ (𝐴 ≥ 𝐵 ↔ 𝐴◡ ≤ 𝐵)
3		brcnvg 5750	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴◡ ≤ 𝐵 ↔ 𝐵 ≤ 𝐴))
4	2, 3	syl5bb 285	1 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ≥ 𝐵 ↔ 𝐵 ≤ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2114  Vcvv 3494   class class class wbr 5066  ^◡ccnv 5554   ≤ cle 10676   ≥ cge-real 44839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-cnv 5563  df-gte 44841

This theorem is referenced by: (None)