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Metamath Proof Explorer

Theorem min1

Description: The minimum of two numbers is less than or equal to the first. (Contributed by NM, 3-Aug-2007)

Ref Expression
Assertion min1 ⊢ A ∈ ℝ ∧ B ∈ ℝ → if A ≤ B A B ≤ A

Proof

Step Hyp Ref Expression
1 rexr ⊢ A ∈ ℝ → A ∈ ℝ *
2 rexr ⊢ B ∈ ℝ → B ∈ ℝ *
3 xrmin1 ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → if A ≤ B A B ≤ A
4 1 2 3 syl2an ⊢ A ∈ ℝ ∧ B ∈ ℝ → if A ≤ B A B ≤ A