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 )