Theorem bj-prex

Description: Existence of unordered pairs proved from ax-bj-sn and ax-bj-bun . (Contributed by BJ, 12-Jan-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-prex	$$⊢\left\{A,B\right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-pr	$$⊢\left\{A,B\right\}=\left\{A\right\}\cup \left\{B\right\}$$
2		bj-snex	$$⊢\left\{A\right\}\in \mathrm{V}$$
3		bj-snex	$$⊢\left\{B\right\}\in \mathrm{V}$$
4		bj-unexg	$$⊢\left(\left\{A\right\}\in \mathrm{V}\wedge \left\{B\right\}\in \mathrm{V}\right)\to \left\{A\right\}\cup \left\{B\right\}\in \mathrm{V}$$
5	2 3 4	mp2an	$$⊢\left\{A\right\}\cup \left\{B\right\}\in \mathrm{V}$$
6	1 5	eqeltri	$$⊢\left\{A,B\right\}\in \mathrm{V}$$