Axiom ax-pre-apti 6778

Description: Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 6749. (Contributed by Jim Kingdon, 29-Jan-2020.)

Assertion

Ref	Expression
ax-pre-apti	⊢ ((A ∈ ℝ ∧ B ∈ ℝ ∧ ¬ (A <ℝ B ∨ B <ℝ A)) → A = B)

Detailed syntax breakdown of Axiom ax-pre-apti

Step	Hyp	Ref	Expression
1		cA	. . . 4 class A
2		cr 6690	. . . 4 class ℝ
3	1, 2	wcel 1390	. . 3 wff A ∈ ℝ
4		cB	. . . 4 class B
5	4, 2	wcel 1390	. . 3 wff B ∈ ℝ
6		cltrr 6695	. . . . . 6 class <ℝ
7	1, 4, 6	wbr 3755	. . . . 5 wff A <ℝ B
8	4, 1, 6	wbr 3755	. . . . 5 wff B <ℝ A
9	7, 8	wo 628	. . . 4 wff (A <ℝ B ∨ B <ℝ A)
10	9	wn 3	. . 3 wff ¬ (A <ℝ B ∨ B <ℝ A)
11	3, 5, 10	w3a 884	. 2 wff (A ∈ ℝ ∧ B ∈ ℝ ∧ ¬ (A <ℝ B ∨ B <ℝ A))
12	1, 4	wceq 1242	. 2 wff A = B
13	11, 12	wi 4	1 wff ((A ∈ ℝ ∧ B ∈ ℝ ∧ ¬ (A <ℝ B ∨ B <ℝ A)) → A = B)

This axiom is referenced by:  axapti  6867