Theorem eqlei

Description: Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999) (Revised by Alexander van der Vekens, 20-Mar-2018)

	Ref	Expression
Hypothesis	lt.1	$$⊢A\in ℝ$$
Assertion	eqlei	$$⊢A=B\to A\le B$$

Proof

Step	Hyp	Ref	Expression
1		lt.1	$$⊢A\in ℝ$$
2		eleq1a	$$⊢A\in ℝ\to \left(B=A\to B\in ℝ\right)$$
3	1 2	ax-mp	$$⊢B=A\to B\in ℝ$$
4	3	eqcoms	$$⊢A=B\to B\in ℝ$$
5		letri3	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A=B\leftrightarrow \left(A\le B\wedge B\le A\right)\right)$$
6	1 5	mpan	$$⊢B\in ℝ\to \left(A=B\leftrightarrow \left(A\le B\wedge B\le A\right)\right)$$
7		simpl	$$⊢\left(A\le B\wedge B\le A\right)\to A\le B$$
8	6 7	biimtrdi	$$⊢B\in ℝ\to \left(A=B\to A\le B\right)$$
9	4 8	mpcom	$$⊢A=B\to A\le B$$