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Metamath Proof Explorer

Theorem xreqle

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

		Ref	Expression
	Assertion	xreqle	$⊢ (A \in ℝ^{*} \land A = B) \to A \leq B$

Step	Hyp	Ref	Expression
1		xrleid	$⊢ A \in ℝ^{*} \to A \leq A$
2	1	adantr	$⊢ (A \in ℝ^{*} \land A = B) \to A \leq A$
3		simpr	$⊢ (A \in ℝ^{*} \land A = B) \to A = B$
4		breq2	$⊢ A = B \to (A \leq A \leftrightarrow A \leq B)$
5	4	biimpac	$⊢ (A \leq A \land A = B) \to A \leq B$
6	2 3 5	syl2anc	$⊢ (A \in ℝ^{*} \land A = B) \to A \leq B$