Theorem xrletrid

Description: Trichotomy law for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	xrletrid.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xrletrid.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
	xrletrid.3	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
	xrletrid.4	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
Assertion	xrletrid	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		xrletrid.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xrletrid.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
3		xrletrid.3	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
4		xrletrid.4	⊢ ( 𝜑 → 𝐵 ≤ 𝐴 )
5		xrletri3	⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ) )
6	1 2 5	syl2anc	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ) )
7	3 4 6	mpbir2and	⊢ ( 𝜑 → 𝐴 = 𝐵 )