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	$$⊢A\setminus A=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		dfdif2	$$⊢A\setminus A=\left\{x\in A|\neg x\in A\right\}$$
2		dfnul3	$$⊢\varnothing =\left\{x\in A|\neg x\in A\right\}$$
3	1 2	eqtr4i	$$⊢A\setminus A=\varnothing$$