Theorem axlttri

Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005)

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

Proof

Step	Hyp	Ref	Expression
1		ax-pre-lttri	|- ( ( A e. RR /\ B e. RR ) -> ( A -. ( A = B \/ B
2		ltxrlt	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A
3		ltxrlt	|- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> B
4	3	ancoms	|- ( ( A e. RR /\ B e. RR ) -> ( B < A <-> B
5	4	orbi2d	|- ( ( A e. RR /\ B e. RR ) -> ( ( A = B \/ B < A ) <-> ( A = B \/ B
6	5	notbid	|- ( ( A e. RR /\ B e. RR ) -> ( -. ( A = B \/ B < A ) <-> -. ( A = B \/ B
7	1 2 6	3bitr4d	|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> -. ( A = B \/ B < A ) ) )