Theorem eqleltd

Description: Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001)

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

Proof

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