Theorem axlttrn

Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. This restates ax-pre-lttrn with ordering on the extended reals. New proofs should use lttr instead for naming consistency. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005)

	Ref	Expression
Assertion	axlttrn	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )

Proof

Step	Hyp	Ref	Expression
1		ax-pre-lttrn	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A A
2		ltxrlt	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A
3	2	3adant3	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A
4		ltxrlt	|- ( ( B e. RR /\ C e. RR ) -> ( B < C <-> B
5	4	3adant1	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B < C <-> B
6	3 5	anbi12d	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) <-> ( A
7		ltxrlt	|- ( ( A e. RR /\ C e. RR ) -> ( A < C <-> A
8	7	3adant2	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> A
9	1 6 8	3imtr4d	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )