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Axiom ax-pre-apti 7755
 Description: Apartness of reals is tight. Axiom for real and complex numbers, justified by theorem axpre-apti 7713. (Contributed by Jim Kingdon, 29-Jan-2020.)
Assertion
Ref Expression
ax-pre-apti ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)

Detailed syntax breakdown of Axiom ax-pre-apti
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 7639 . . . 4 class
31, 2wcel 1481 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1481 . . 3 wff 𝐵 ∈ ℝ
6 cltrr 7644 . . . . . 6 class <
71, 4, 6wbr 3933 . . . . 5 wff 𝐴 < 𝐵
84, 1, 6wbr 3933 . . . . 5 wff 𝐵 < 𝐴
97, 8wo 698 . . . 4 wff (𝐴 < 𝐵𝐵 < 𝐴)
109wn 3 . . 3 wff ¬ (𝐴 < 𝐵𝐵 < 𝐴)
113, 5, 10w3a 963 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴))
121, 4wceq 1332 . 2 wff 𝐴 = 𝐵
1311, 12wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 < 𝐵𝐵 < 𝐴)) → 𝐴 = 𝐵)
 Colors of variables: wff set class This axiom is referenced by:  axapti  7855
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