Theorem eqoreldif

Description: An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020) (Proof shortened by JJ, 23-Jul-2021)

	Ref	Expression
Assertion	eqoreldif	⊢ ( 𝐵 ∈ 𝐶 → ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 = 𝐵 ∨ 𝐴 ∈ ( 𝐶 ∖ { 𝐵 } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 = 𝐵 ) → 𝐴 ∈ 𝐶 )
2		elsni	⊢ ( 𝐴 ∈ { 𝐵 } → 𝐴 = 𝐵 )
3	2	con3i	⊢ ( ¬ 𝐴 = 𝐵 → ¬ 𝐴 ∈ { 𝐵 } )
4	3	adantl	⊢ ( ( 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐴 ∈ { 𝐵 } )
5	1 4	eldifd	⊢ ( ( 𝐴 ∈ 𝐶 ∧ ¬ 𝐴 = 𝐵 ) → 𝐴 ∈ ( 𝐶 ∖ { 𝐵 } ) )
6	5	ex	⊢ ( 𝐴 ∈ 𝐶 → ( ¬ 𝐴 = 𝐵 → 𝐴 ∈ ( 𝐶 ∖ { 𝐵 } ) ) )
7	6	orrd	⊢ ( 𝐴 ∈ 𝐶 → ( 𝐴 = 𝐵 ∨ 𝐴 ∈ ( 𝐶 ∖ { 𝐵 } ) ) )
8		eleq1a	⊢ ( 𝐵 ∈ 𝐶 → ( 𝐴 = 𝐵 → 𝐴 ∈ 𝐶 ) )
9		eldifi	⊢ ( 𝐴 ∈ ( 𝐶 ∖ { 𝐵 } ) → 𝐴 ∈ 𝐶 )
10	9	a1i	⊢ ( 𝐵 ∈ 𝐶 → ( 𝐴 ∈ ( 𝐶 ∖ { 𝐵 } ) → 𝐴 ∈ 𝐶 ) )
11	8 10	jaod	⊢ ( 𝐵 ∈ 𝐶 → ( ( 𝐴 = 𝐵 ∨ 𝐴 ∈ ( 𝐶 ∖ { 𝐵 } ) ) → 𝐴 ∈ 𝐶 ) )
12	7 11	impbid2	⊢ ( 𝐵 ∈ 𝐶 → ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 = 𝐵 ∨ 𝐴 ∈ ( 𝐶 ∖ { 𝐵 } ) ) ) )