Theorems · Theorem · number theory
exists_irrational_btwn
∀ {x y : ℝ}, x < y → ∃ r, Irrational r ∧ x < r ∧ r < yThere is an irrational number r between any two reals x < r < y.
- Defined in
- Mathlib.NumberTheory.Real.Irrational
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 134 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Real.sqrtproof · cited by 545
- Irrationalstatement and proof · cited by 139
- sub_lt_iff_lt_addproof · cited by 39
- exists_rat_btwnproof · cited by 26
- lt_sub_iff_add_ltproof · cited by 24
- sub_lt_sub_iff_rightproof · cited by 8
- Irrational.ratCast_addproof · cited by 6
- irrational_sqrt_twoproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- dense_irrationalproof · cited by 2