Theorem unab

Description: Union of two class abstractions. (Contributed by NM, 29-Sep-2002) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	unab	$$⊢\left\{x|\phi \right\}\cup \left\{x|\psi \right\}=\left\{x|\left(\phi \vee \psi \right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		sbor	$$⊢\left[y/x\right]\left(\phi \vee \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \vee \left[y/x\right]\psi \right)$$
2		df-clab	$$⊢y\in \left\{x|\left(\phi \vee \psi \right)\right\}\leftrightarrow \left[y/x\right]\left(\phi \vee \psi \right)$$
3		df-clab	$$⊢y\in \left\{x|\phi \right\}\leftrightarrow \left[y/x\right]\phi$$
4		df-clab	$$⊢y\in \left\{x|\psi \right\}\leftrightarrow \left[y/x\right]\psi$$
5	3 4	orbi12i	$$⊢\left(y\in \left\{x|\phi \right\}\vee y\in \left\{x|\psi \right\}\right)\leftrightarrow \left(\left[y/x\right]\phi \vee \left[y/x\right]\psi \right)$$
6	1 2 5	3bitr4ri	$$⊢\left(y\in \left\{x|\phi \right\}\vee y\in \left\{x|\psi \right\}\right)\leftrightarrow y\in \left\{x|\left(\phi \vee \psi \right)\right\}$$
7	6	uneqri	$$⊢\left\{x|\phi \right\}\cup \left\{x|\psi \right\}=\left\{x|\left(\phi \vee \psi \right)\right\}$$