Theorem ltled

Description: 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltd.2	$$⊢\phi \to B\in ℝ$$
	ltled.1	$$⊢\phi \to A<B$$
Assertion	ltled	$$⊢\phi \to A\le B$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		ltled.1	$$⊢\phi \to A<B$$
4		ltle	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\to A\le B\right)$$
5	1 2 4	syl2anc	$$⊢\phi \to \left(A<B\to A\le B\right)$$
6	3 5	mpd	$$⊢\phi \to A\le B$$