Theorem leltned

Description: 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltd.2	$$⊢\phi \to B\in ℝ$$
	leltned.3	$$⊢\phi \to A\le B$$
Assertion	leltned	$$⊢\phi \to \left(A<B\leftrightarrow B\ne A\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		leltned.3	$$⊢\phi \to A\le B$$
4		leltne	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge A\le B\right)\to \left(A<B\leftrightarrow B\ne A\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A<B\leftrightarrow B\ne A\right)$$