Theorem unjust

Description: Soundness justification theorem for df-un . (Contributed by Rodolfo Medina, 28-Apr-2010) (Proof shortened by Andrew Salmon, 9-Jul-2011)

	Ref	Expression
Assertion	unjust	$$⊢\left\{x|\left(x\in A\vee x\in B\right)\right\}=\left\{y|\left(y\in A\vee y\in B\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		eleq1w	$$⊢x=z\to \left(x\in A\leftrightarrow z\in A\right)$$
2		eleq1w	$$⊢x=z\to \left(x\in B\leftrightarrow z\in B\right)$$
3	1 2	orbi12d	$$⊢x=z\to \left(\left(x\in A\vee x\in B\right)\leftrightarrow \left(z\in A\vee z\in B\right)\right)$$
4	3	cbvabv	$$⊢\left\{x|\left(x\in A\vee x\in B\right)\right\}=\left\{z|\left(z\in A\vee z\in B\right)\right\}$$
5		eleq1w	$$⊢z=y\to \left(z\in A\leftrightarrow y\in A\right)$$
6		eleq1w	$$⊢z=y\to \left(z\in B\leftrightarrow y\in B\right)$$
7	5 6	orbi12d	$$⊢z=y\to \left(\left(z\in A\vee z\in B\right)\leftrightarrow \left(y\in A\vee y\in B\right)\right)$$
8	7	cbvabv	$$⊢\left\{z|\left(z\in A\vee z\in B\right)\right\}=\left\{y|\left(y\in A\vee y\in B\right)\right\}$$
9	4 8	eqtri	$$⊢\left\{x|\left(x\in A\vee x\in B\right)\right\}=\left\{y|\left(y\in A\vee y\in B\right)\right\}$$