This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem max2

Description: A number is less than or equal to the maximum of it and another. (Contributed by NM, 3-Apr-2005)

Ref Expression
Assertion max2 ⊢ A ∈ ℝ ∧ B ∈ ℝ → B ≤ if A ≤ B B A

Proof

Step Hyp Ref Expression
1 rexr ⊢ A ∈ ℝ → A ∈ ℝ *
2 rexr ⊢ B ∈ ℝ → B ∈ ℝ *
3 xrmax2 ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → B ≤ if A ≤ B B A
4 1 2 3 syl2an ⊢ A ∈ ℝ ∧ B ∈ ℝ → B ≤ if A ≤ B B A