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Metamath Proof Explorer

Theorem xrltnle

Description: "Less than" expressed in terms of "less than or equal to", for extended reals. (Contributed by NM, 6-Feb-2007)

Ref Expression
Assertion xrltnle ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → A < B ↔ ¬ B ≤ A

Proof

Step Hyp Ref Expression
1 xrlenlt ⊢ B ∈ ℝ * ∧ A ∈ ℝ * → B ≤ A ↔ ¬ A < B
2 1 con2bid ⊢ B ∈ ℝ * ∧ A ∈ ℝ * → A < B ↔ ¬ B ≤ A
3 2 ancoms ⊢ A ∈ ℝ * ∧ B ∈ ℝ * → A < B ↔ ¬ B ≤ A