Axiom ax-pre-ltwlin 8009

Description: Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 7967. (Contributed by Jim Kingdon, 12-Jan-2020.)

Assertion

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

Detailed syntax breakdown of Axiom ax-pre-ltwlin

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cr 7895	. . . 4 class RR
3	1, 2	wcel 2167	. . 3 wff A e. RR
4		cB	. . . 4 class B
5	4, 2	wcel 2167	. . 3 wff B e. RR
6		cC	. . . 4 class C
7	6, 2	wcel 2167	. . 3 wff C e. RR
8	3, 5, 7	w3a 980	. 2 wff ( A e. RR /\ B e. RR /\ C e. RR )
9		cltrr 7900	. . . 4 class <RR
10	1, 4, 9	wbr 4034	. . 3 wff A <RR B
11	1, 6, 9	wbr 4034	. . . 4 wff A <RR C
12	6, 4, 9	wbr 4034	. . . 4 wff C <RR B
13	11, 12	wo 709	. . 3 wff ( A <RR C \/ C <RR B )
14	10, 13	wi 4	. 2 wff ( A <RR B -> ( A <RR C \/ C <RR B ) )
15	8, 14	wi 4	1 wff ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( A <RR C \/ C <RR B ) ) )

This axiom is referenced by:  axltwlin  8111