Theorem lenlteq

Description: 'less than or equal to' but not 'less than' implies 'equal' . (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	lenlteq.1	$$⊢\phi \to A\in ℝ$$
	lenlteq.2	$$⊢\phi \to B\in ℝ$$
	lenlteq.3	$$⊢\phi \to A\le B$$
	lenlteq.4	$$⊢\phi \to \neg A<B$$
Assertion	lenlteq	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		lenlteq.1	$$⊢\phi \to A\in ℝ$$
2		lenlteq.2	$$⊢\phi \to B\in ℝ$$
3		lenlteq.3	$$⊢\phi \to A\le B$$
4		lenlteq.4	$$⊢\phi \to \neg A<B$$
5	3 4	jca	$$⊢\phi \to \left(A\le B\wedge \neg A<B\right)$$
6		eqlelt	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(A=B\leftrightarrow \left(A\le B\wedge \neg A<B\right)\right)$$
7	1 2 6	syl2anc	$$⊢\phi \to \left(A=B\leftrightarrow \left(A\le B\wedge \neg A<B\right)\right)$$
8	5 7	mpbird	$$⊢\phi \to A=B$$