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) (Revised by David Abernethy, 17-Jun-2012)

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

Proof

Step	Hyp	Ref	Expression
1		dfdif2	⊢ ( 𝐴 ∖ 𝐴 ) = { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴 }
2		dfnul3	⊢ ∅ = { 𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴 }
3	1 2	eqtr4i	⊢ ( 𝐴 ∖ 𝐴 ) = ∅