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

Metamath Proof Explorer

Theorem eqsnd

Description: Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023) (Proof shortened by SN, 3-Jul-2025)

Ref Expression
Hypotheses eqsnd.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 = 𝐵 )
eqsnd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
Assertion eqsnd ( 𝜑 → 𝐴 = { 𝐵 } )

Proof

Step Hyp Ref Expression
1 eqsnd.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 = 𝐵 )
2 eqsnd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
3 1 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝑥 = 𝐵 )
4 2 ne0d ⊢ ( 𝜑 → 𝐴 ≠ ∅ )
5 eqsn ⊢ ( 𝐴 ≠ ∅ → ( 𝐴 = { 𝐵 } ↔ ∀ 𝑥 ∈ 𝐴 𝑥 = 𝐵 ) )
6 4 5 syl ⊢ ( 𝜑 → ( 𝐴 = { 𝐵 } ↔ ∀ 𝑥 ∈ 𝐴 𝑥 = 𝐵 ) )
7 3 6 mpbird ⊢ ( 𝜑 → 𝐴 = { 𝐵 } )