Theorem ltlend

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

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

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		ltlen	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A<B\leftrightarrow \left(A\le B\wedge B\ne A\right)\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \left(A<B\leftrightarrow \left(A\le B\wedge B\ne A\right)\right)$$