Theorem ltnsymd

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

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ltd.2	$$⊢\phi \to B\in ℝ$$
	ltled.1	$$⊢\phi \to A<B$$
Assertion	ltnsymd	$$⊢\phi \to \neg B<A$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ltd.2	$$⊢\phi \to B\in ℝ$$
3		ltled.1	$$⊢\phi \to A<B$$
4	1 2 3	ltled	$$⊢\phi \to A\le B$$
5	1 2	lenltd	$$⊢\phi \to \left(A\le B\leftrightarrow \neg B<A\right)$$
6	4 5	mpbid	$$⊢\phi \to \neg B<A$$