Axiom ax-pre-lttri 10010

Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 9986. Note: The more general version for extended reals is axlttri 10109. Normally new proofs would use xrlttri 11972. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
ax-pre-lttri	|- ( ( A e. RR /\ B e. RR ) -> ( A <RR B <-> -. ( A = B \/ B <RR A ) ) )

Detailed syntax breakdown of Axiom ax-pre-lttri

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cr 9935	. . . 4 class RR
3	1, 2	wcel 1990	. . 3 wff A e. RR
4		cB	. . . 4 class B
5	4, 2	wcel 1990	. . 3 wff B e. RR
6	3, 5	wa 384	. 2 wff ( A e. RR /\ B e. RR )
7		cltrr 9940	. . . 4 class <RR
8	1, 4, 7	wbr 4653	. . 3 wff A <RR B
9	1, 4	wceq 1483	. . . . 5 wff A = B
10	4, 1, 7	wbr 4653	. . . . 5 wff B <RR A
11	9, 10	wo 383	. . . 4 wff ( A = B \/ B <RR A )
12	11	wn 3	. . 3 wff -. ( A = B \/ B <RR A )
13	8, 12	wb 196	. 2 wff ( A <RR B <-> -. ( A = B \/ B <RR A ) )
14	6, 13	wi 4	1 wff ( ( A e. RR /\ B e. RR ) -> ( A <RR B <-> -. ( A = B \/ B <RR A ) ) )

This axiom is referenced by:  axlttri  10109