Theorem iccdisj2

Description: If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024)

	Ref	Expression
Assertion	iccdisj2	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> ( ( A [,] B ) i^i ( C [,] D ) ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		simp1	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> A e. RR* )
2		simp3	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> B < C )
3		ltrelxr	|- < C_ ( RR* X. RR* )
4	3	brel	|- ( B < C -> ( B e. RR* /\ C e. RR* ) )
5	2 4	syl	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> ( B e. RR* /\ C e. RR* ) )
6	5	simprd	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> C e. RR* )
7	1	xrleidd	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> A <_ A )
8		iccssico	|- ( ( ( A e. RR* /\ C e. RR* ) /\ ( A <_ A /\ B < C ) ) -> ( A [,] B ) C_ ( A [,) C ) )
9	1 6 7 2 8	syl22anc	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> ( A [,] B ) C_ ( A [,) C ) )
10		simp2	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> D e. RR* )
11		df-ico	|- [,) = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z < y ) } )
12		df-icc	|- [,] = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x <_ z /\ z <_ y ) } )
13		xrlenlt	|- ( ( C e. RR* /\ w e. RR* ) -> ( C <_ w <-> -. w < C ) )
14	11 12 13	ixxdisj	|- ( ( A e. RR* /\ C e. RR* /\ D e. RR* ) -> ( ( A [,) C ) i^i ( C [,] D ) ) = (/) )
15	1 6 10 14	syl3anc	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> ( ( A [,) C ) i^i ( C [,] D ) ) = (/) )
16	9 15	ssdisjd	|- ( ( A e. RR* /\ D e. RR* /\ B < C ) -> ( ( A [,] B ) i^i ( C [,] D ) ) = (/) )