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 ∈ 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 7895	. . . 4 class ℝ
3	1, 2	wcel 2167	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2167	. . 3 wff 𝐵 ∈ ℝ
6		cltrr 7900	. . . . . 6 class <ℝ
7	1, 4, 6	wbr 4034	. . . . 5 wff 𝐴 <ℝ 𝐵
8	4, 1, 6	wbr 4034	. . . . 5 wff 𝐵 <ℝ 𝐴
9	7, 8	wo 709	. . . 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 980	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))
12	1, 4	wceq 1364	. 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  8114