Theorem difid

Description: The difference between a class and itself is the empty set. Proposition 5.15 of TakeutiZaring p. 20. Also Theorem 32 of Suppes p. 28. (Contributed by NM, 22-Apr-2004)

	Ref	Expression
Assertion	difid	⊢ ( 𝐴 ∖ 𝐴 ) = ∅

Proof

Step	Hyp	Ref	Expression
1		ssid	⊢ 𝐴 ⊆ 𝐴
2		ssdif0	⊢ ( 𝐴 ⊆ 𝐴 ↔ ( 𝐴 ∖ 𝐴 ) = ∅ )
3	1 2	mpbi	⊢ ( 𝐴 ∖ 𝐴 ) = ∅