Theorem eldifsnbd

Description: Membership in a set with an element removed implies non-equality with that element. (Contributed by Thierry Arnoux, 13-Jul-2026)

	Ref	Expression
Hypothesis	eldifsnbd.1	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 ∖ { 𝐶 } ) )
Assertion	eldifsnbd	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eldifsnbd.1	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 ∖ { 𝐶 } ) )
2		eldifsn	⊢ ( 𝐴 ∈ ( 𝐵 ∖ { 𝐶 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ) )
3	1 2	sylib	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ) )
4	3	simprd	⊢ ( 𝜑 → 𝐴 ≠ 𝐶 )