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	⊢ { 𝐴 , 𝐵 } ∈ V

Proof

Step	Hyp	Ref	Expression
1		df-pr	⊢ { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )
2		bj-snex	⊢ { 𝐴 } ∈ V
3		bj-snex	⊢ { 𝐵 } ∈ V
4		bj-unexg	⊢ ( ( { 𝐴 } ∈ V ∧ { 𝐵 } ∈ V ) → ( { 𝐴 } ∪ { 𝐵 } ) ∈ V )
5	2 3 4	mp2an	⊢ ( { 𝐴 } ∪ { 𝐵 } ) ∈ V
6	1 5	eqeltri	⊢ { 𝐴 , 𝐵 } ∈ V