Theorem lensymd

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

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

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		lensymd.3	$$⊢\phi \to A\le B$$
4	1 2	lenltd	$$⊢\phi \to \left(A\le B\leftrightarrow \neg B<A\right)$$
5	3 4	mpbid	$$⊢\phi \to \neg B<A$$