Description: Consequence of trichotomy. (Contributed by NM, 14-May-1999)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lt.1 | ⊢ 𝐴 ∈ ℝ | |
lt.2 | ⊢ 𝐵 ∈ ℝ | ||
Assertion | letri3i | ⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt.1 | ⊢ 𝐴 ∈ ℝ | |
2 | lt.2 | ⊢ 𝐵 ∈ ℝ | |
3 | letri3 | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ) ) | |
4 | 1 2 3 | mp2an | ⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ) |