Theorem eldifsni

Description: Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015)

	Ref	Expression
Assertion	eldifsni	⊢ ( 𝐴 ∈ ( 𝐵 ∖ { 𝐶 } ) → 𝐴 ≠ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eldifsn	⊢ ( 𝐴 ∈ ( 𝐵 ∖ { 𝐶 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ) )
2	1	simprbi	⊢ ( 𝐴 ∈ ( 𝐵 ∖ { 𝐶 } ) → 𝐴 ≠ 𝐶 )