Metamath Proof Explorer

Theorem xreqle

Description: Equality implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Assertion xreqle ( ( 𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵 ) → 𝐴 ≤ 𝐵 )

Proof

Step Hyp Ref Expression
1 xrleid ⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 )
2 1 adantr ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵 ) → 𝐴 ≤ 𝐴 )
3 simpr ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 )
4 breq2 ⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≤ 𝐴 ↔ 𝐴 ≤ 𝐵 ) )
5 4 biimpac ⊢ ( ( 𝐴 ≤ 𝐴 ∧ 𝐴 = 𝐵 ) → 𝐴 ≤ 𝐵 )
6 2 3 5 syl2anc ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵 ) → 𝐴 ≤ 𝐵 )