Axiom ax-pre-lttrn 10011

Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 9987. Note: The more general version for extended reals is axlttrn 10110. Normally new proofs would use lttr 10114. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)

Assertion

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

Detailed syntax breakdown of Axiom ax-pre-lttrn

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		cC	. . . 4 class C
7	6, 2	wcel 1990	. . 3 wff C e. RR
8	3, 5, 7	w3a 1037	. 2 wff ( A e. RR /\ B e. RR /\ C e. RR )
9		cltrr 9940	. . . . 5 class <RR
10	1, 4, 9	wbr 4653	. . . 4 wff A <RR B
11	4, 6, 9	wbr 4653	. . . 4 wff B <RR C
12	10, 11	wa 384	. . 3 wff ( A <RR B /\ B <RR C )
13	1, 6, 9	wbr 4653	. . 3 wff A <RR C
14	12, 13	wi 4	. 2 wff ( ( A <RR B /\ B <RR C ) -> A <RR C )
15	8, 14	wi 4	1 wff ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <RR B /\ B <RR C ) -> A <RR C ) )

This axiom is referenced by:  axlttrn  10110