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Mirrors > Home > ILE Home > Th. List > ax-pre-apti | GIF version |
Description: Apartness of reals is tight. Axiom for real and complex numbers, justified by Theorem axpre-apti 7847. (Contributed by Jim Kingdon, 29-Jan-2020.) |
Ref | Expression |
---|---|
ax-pre-apti | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | cr 7773 | . . . 4 class ℝ | |
3 | 1, 2 | wcel 2141 | . . 3 wff 𝐴 ∈ ℝ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2141 | . . 3 wff 𝐵 ∈ ℝ |
6 | cltrr 7778 | . . . . . 6 class <ℝ | |
7 | 1, 4, 6 | wbr 3989 | . . . . 5 wff 𝐴 <ℝ 𝐵 |
8 | 4, 1, 6 | wbr 3989 | . . . . 5 wff 𝐵 <ℝ 𝐴 |
9 | 7, 8 | wo 703 | . . . 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 973 | . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) |
12 | 1, 4 | wceq 1348 | . 2 wff 𝐴 = 𝐵 |
13 | 11, 12 | wi 4 | 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵) |
Colors of variables: wff set class |
This axiom is referenced by: axapti 7990 |
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