Theorem inundif

Description: The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	inundif	$$⊢\left(A\cap B\right)\cup \left(A\setminus B\right)=A$$

Proof

Step	Hyp	Ref	Expression
1		elin	$$⊢x\in \left(A\cap B\right)\leftrightarrow \left(x\in A\wedge x\in B\right)$$
2		eldif	$$⊢x\in \left(A\setminus B\right)\leftrightarrow \left(x\in A\wedge \neg x\in B\right)$$
3	1 2	orbi12i	$$⊢\left(x\in \left(A\cap B\right)\vee x\in \left(A\setminus B\right)\right)\leftrightarrow \left(\left(x\in A\wedge x\in B\right)\vee \left(x\in A\wedge \neg x\in B\right)\right)$$
4		pm4.42	$$⊢x\in A\leftrightarrow \left(\left(x\in A\wedge x\in B\right)\vee \left(x\in A\wedge \neg x\in B\right)\right)$$
5	3 4	bitr4i	$$⊢\left(x\in \left(A\cap B\right)\vee x\in \left(A\setminus B\right)\right)\leftrightarrow x\in A$$
6	5	uneqri	$$⊢\left(A\cap B\right)\cup \left(A\setminus B\right)=A$$