Theorem eldifsn

Description: Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007)

	Ref	Expression
Assertion	eldifsn	⊢ ( 𝐴 ∈ ( 𝐵 ∖ { 𝐶 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		eldif	⊢ ( 𝐴 ∈ ( 𝐵 ∖ { 𝐶 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ { 𝐶 } ) )
2		elsng	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ { 𝐶 } ↔ 𝐴 = 𝐶 ) )
3	2	necon3bbid	⊢ ( 𝐴 ∈ 𝐵 → ( ¬ 𝐴 ∈ { 𝐶 } ↔ 𝐴 ≠ 𝐶 ) )
4	3	pm5.32i	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ { 𝐶 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ) )
5	1 4	bitri	⊢ ( 𝐴 ∈ ( 𝐵 ∖ { 𝐶 } ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶 ) )