Theorem leneltd

Description: 'Less than or equal to' and 'not equals' implies 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltd.2	$$⊢\phi \to B\in ℝ$$
	leltned.3	$$⊢\phi \to A\le B$$
	leneltd.4	$$⊢\phi \to B\ne A$$
Assertion	leneltd	$$⊢\phi \to A<B$$

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		leneltd.4	$$⊢\phi \to B\ne A$$
5	1 2 3	leltned	$$⊢\phi \to \left(A<B\leftrightarrow B\ne A\right)$$
6	4 5	mpbird	$$⊢\phi \to A<B$$