Theorem ltlecasei

Description: Ordering elimination by cases. (Contributed by NM, 1-Jul-2007) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltlecasei.1	|- ( ( ph /\ A < B ) -> ps )
	ltlecasei.2	|- ( ( ph /\ B <_ A ) -> ps )
	ltlecasei.3	|- ( ph -> A e. RR )
	ltlecasei.4	|- ( ph -> B e. RR )
Assertion	ltlecasei	|- ( ph -> ps )

Proof

Step	Hyp	Ref	Expression
1		ltlecasei.1	|- ( ( ph /\ A < B ) -> ps )
2		ltlecasei.2	|- ( ( ph /\ B <_ A ) -> ps )
3		ltlecasei.3	|- ( ph -> A e. RR )
4		ltlecasei.4	|- ( ph -> B e. RR )
5		lelttric	|- ( ( B e. RR /\ A e. RR ) -> ( B <_ A \/ A < B ) )
6	4 3 5	syl2anc	|- ( ph -> ( B <_ A \/ A < B ) )
7	2 1 6	mpjaodan	|- ( ph -> ps )