Theorem unidom

Description: An upper bound for the cardinality of a union. Theorem 10.47 of TakeutiZaring p. 98. (Contributed by NM, 25-Mar-2006) (Proof shortened by Mario Carneiro, 1-Sep-2015)

	Ref	Expression
Assertion	unidom	$$⊢\left(A\in V\wedge \forall x\in Ax\preccurlyeq B\right)\to \bigcup A\preccurlyeq \left(A\times B\right)$$

Proof

Step	Hyp	Ref	Expression
1		uniiun	$$⊢\bigcup A=\bigcup _{x\in A}x$$
2		iundom	$$⊢\left(A\in V\wedge \forall x\in Ax\preccurlyeq B\right)\to \bigcup _{x\in A}x\preccurlyeq \left(A\times B\right)$$
3	1 2	eqbrtrid	$$⊢\left(A\in V\wedge \forall x\in Ax\preccurlyeq B\right)\to \bigcup A\preccurlyeq \left(A\times B\right)$$