Description: Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | reutru | ⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑥 ∈ 𝐴 ⊤ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru | ⊢ ⊤ | |
2 | 1 | biantru | ⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ ⊤ ) ) |
3 | 2 | eubii | ⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ⊤ ) ) |
4 | df-reu | ⊢ ( ∃! 𝑥 ∈ 𝐴 ⊤ ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ⊤ ) ) | |
5 | 3 4 | bitr4i | ⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑥 ∈ 𝐴 ⊤ ) |