Axiom ax-pre-apti 8011

Description: Apartness of reals is tight. Axiom for real and complex numbers, justified by Theorem axpre-apti 7969. (Contributed by Jim Kingdon, 29-Jan-2020.)

Assertion

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

Detailed syntax breakdown of Axiom ax-pre-apti

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		cltrr 7900	. . . . . 6 class <RR
7	1, 4, 6	wbr 4034	. . . . 5 wff A <RR B
8	4, 1, 6	wbr 4034	. . . . 5 wff B <RR A
9	7, 8	wo 709	. . . 4 wff ( A <RR B \/ B <RR A )
10	9	wn 3	. . 3 wff -. ( A <RR B \/ B <RR A )
11	3, 5, 10	w3a 980	. 2 wff ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) )
12	1, 4	wceq 1364	. 2 wff A = B
13	11, 12	wi 4	1 wff ( ( A e. RR /\ B e. RR /\ -. ( A <RR B \/ B <RR A ) ) -> A = B )

This axiom is referenced by:  axapti  8114