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	$$⊢\phi \to A\in \left(B\setminus \left\{C\right\}\right)$$
Assertion	eldifsnbd	$$⊢\phi \to A\ne C$$

Proof

Step	Hyp	Ref	Expression
1		eldifsnbd.1	$$⊢\phi \to A\in \left(B\setminus \left\{C\right\}\right)$$
2		eldifsn	$$⊢A\in \left(B\setminus \left\{C\right\}\right)\leftrightarrow \left(A\in B\wedge A\ne C\right)$$
3	1 2	sylib	$$⊢\phi \to \left(A\in B\wedge A\ne C\right)$$
4	3	simprd	$$⊢\phi \to A\ne C$$