Theorem exdifsn

Description: There exists an element in a class excluding a singleton if and only if there exists an element in the original class not equal to the singleton element. (Contributed by BTernaryTau, 15-Sep-2023)

	Ref	Expression
Assertion	exdifsn	⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ↔ ∃ 𝑥 ∈ 𝐴 𝑥 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		eldifsn	⊢ ( 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ) )
2	1	exbii	⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ) )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑥 ≠ 𝐵 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ) )
4	2 3	bitr4i	⊢ ( ∃ 𝑥 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ↔ ∃ 𝑥 ∈ 𝐴 𝑥 ≠ 𝐵 )