Theorem bj-axadj

Description: Two ways of stating the axiom of adjunction (which is the universal closure of either side, see ax-bj-adj ). (Contributed by BJ, 12-Jan-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-axadj	|- ( ( x u. { y } ) e. _V <-> E. z A. t ( t e. z <-> ( t e. x \/ t = y ) ) )

Proof

Step	Hyp	Ref	Expression
1		elun	|- ( t e. ( x u. { y } ) <-> ( t e. x \/ t e. { y } ) )
2		velsn	|- ( t e. { y } <-> t = y )
3	2	orbi2i	|- ( ( t e. x \/ t e. { y } ) <-> ( t e. x \/ t = y ) )
4	1 3	bitri	|- ( t e. ( x u. { y } ) <-> ( t e. x \/ t = y ) )
5	4	bj-clex	|- ( ( x u. { y } ) e. _V <-> E. z A. t ( t e. z <-> ( t e. x \/ t = y ) ) )