Theorem ecun

Description: The union coset of A . (Contributed by Peter Mazsa, 28-Jan-2026)

	Ref	Expression
Assertion	ecun	$$⊢A\in V\to \left[A\right]\left(R\cup S\right)=\left[A\right]R\cup \left[A\right]S$$

Proof

Step	Hyp	Ref	Expression
1		unab	$$⊢\left\{x|ARx\right\}\cup \left\{x|ASx\right\}=\left\{x|\left(ARx\vee ASx\right)\right\}$$
2	1	a1i	$$⊢A\in V\to \left\{x|ARx\right\}\cup \left\{x|ASx\right\}=\left\{x|\left(ARx\vee ASx\right)\right\}$$
3		dfec2	$$⊢A\in V\to \left[A\right]R=\left\{x|ARx\right\}$$
4		dfec2	$$⊢A\in V\to \left[A\right]S=\left\{x|ASx\right\}$$
5	3 4	uneq12d	$$⊢A\in V\to \left[A\right]R\cup \left[A\right]S=\left\{x|ARx\right\}\cup \left\{x|ASx\right\}$$
6		elecALTV	$$⊢\left(A\in V\wedge x\in \mathrm{V}\right)\to \left(x\in \left[A\right]\left(R\cup S\right)\leftrightarrow A\left(R\cup S\right)x\right)$$
7	6	elvd	$$⊢A\in V\to \left(x\in \left[A\right]\left(R\cup S\right)\leftrightarrow A\left(R\cup S\right)x\right)$$
8		brun	$$⊢A\left(R\cup S\right)x\leftrightarrow \left(ARx\vee ASx\right)$$
9	7 8	bitrdi	$$⊢A\in V\to \left(x\in \left[A\right]\left(R\cup S\right)\leftrightarrow \left(ARx\vee ASx\right)\right)$$
10	9	eqabdv	$$⊢A\in V\to \left[A\right]\left(R\cup S\right)=\left\{x|\left(ARx\vee ASx\right)\right\}$$
11	2 5 10	3eqtr4rd	$$⊢A\in V\to \left[A\right]\left(R\cup S\right)=\left[A\right]R\cup \left[A\right]S$$