This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ltlecasei

Description: Ordering elimination by cases. (Contributed by NM, 1-Jul-2007) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses ltlecasei.1 ⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝜓 )
ltlecasei.2 ⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝜓 )
ltlecasei.3 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltlecasei.4 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion ltlecasei ( 𝜑 → 𝜓 )

Proof

Step Hyp Ref Expression
1 ltlecasei.1 ⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝜓 )
2 ltlecasei.2 ⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝜓 )
3 ltlecasei.3 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
4 ltlecasei.4 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
5 lelttric ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝐵 ≤ 𝐴 ∨ 𝐴 < 𝐵 ) )
6 4 3 5 syl2anc ⊢ ( 𝜑 → ( 𝐵 ≤ 𝐴 ∨ 𝐴 < 𝐵 ) )
7 2 1 6 mpjaodan ⊢ ( 𝜑 → 𝜓 )