Theorem unipr

Description: The union of a pair is the union of its members. Proposition 5.7 of TakeutiZaring p. 16. (Contributed by NM, 23-Aug-1993) (Proof shortened by BJ, 1-Sep-2024)

	Ref	Expression
Hypotheses	unipr.1	$$⊢A\in \mathrm{V}$$
	unipr.2	$$⊢B\in \mathrm{V}$$
Assertion	unipr	$$⊢\bigcup \left\{A,B\right\}=A\cup B$$

Proof

Step	Hyp	Ref	Expression
1		unipr.1	$$⊢A\in \mathrm{V}$$
2		unipr.2	$$⊢B\in \mathrm{V}$$
3		uniprg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \bigcup \left\{A,B\right\}=A\cup B$$
4	1 2 3	mp2an	$$⊢\bigcup \left\{A,B\right\}=A\cup B$$