Theorem bj-prexg

Description: Existence of unordered pairs formed on sets, proved from ax-bj-sn and ax-bj-bun . Contrary to bj-prex , this proof is intuitionistically valid and does not require ax-nul . (Contributed by BJ, 12-Jan-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-prexg	$$⊢\left(A\in V\wedge B\in W\right)\to \left\{A,B\right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		df-pr	$$⊢\left\{A,B\right\}=\left\{A\right\}\cup \left\{B\right\}$$
2		bj-snexg	$$⊢A\in V\to \left\{A\right\}\in \mathrm{V}$$
3		bj-snexg	$$⊢B\in W\to \left\{B\right\}\in \mathrm{V}$$
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	syl2an	$$⊢\left(A\in V\wedge B\in W\right)\to \left\{A\right\}\cup \left\{B\right\}\in \mathrm{V}$$
6	1 5	eqeltrid	$$⊢\left(A\in V\wedge B\in W\right)\to \left\{A,B\right\}\in \mathrm{V}$$