![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6769. (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 6710 | . . . 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 6715 | . . . . . 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) |
Colors of variables: wff set class |
This axiom is referenced by: axapti 6887 |
Copyright terms: Public domain | W3C validator |