Theorem sucdifsn

Description: The difference between the successor and the singleton of a class is the class. (Contributed by Peter Mazsa, 20-Sep-2024)

	Ref	Expression
Assertion	sucdifsn	⊢ ( suc 𝐴 ∖ { 𝐴 } ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		df-suc	⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } )
2	1	difeq1i	⊢ ( suc 𝐴 ∖ { 𝐴 } ) = ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } )
3		sucdifsn2	⊢ ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) = 𝐴
4	2 3	eqtri	⊢ ( suc 𝐴 ∖ { 𝐴 } ) = 𝐴