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 ∪ { 𝑦 } ) ∈ V ↔ ∃ 𝑧 ∀ 𝑡 ( 𝑡 ∈ 𝑧 ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elun	⊢ ( 𝑡 ∈ ( 𝑥 ∪ { 𝑦 } ) ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 ∈ { 𝑦 } ) )
2		velsn	⊢ ( 𝑡 ∈ { 𝑦 } ↔ 𝑡 = 𝑦 )
3	2	orbi2i	⊢ ( ( 𝑡 ∈ 𝑥 ∨ 𝑡 ∈ { 𝑦 } ) ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦 ) )
4	1 3	bitri	⊢ ( 𝑡 ∈ ( 𝑥 ∪ { 𝑦 } ) ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦 ) )
5	4	bj-clex	⊢ ( ( 𝑥 ∪ { 𝑦 } ) ∈ V ↔ ∃ 𝑧 ∀ 𝑡 ( 𝑡 ∈ 𝑧 ↔ ( 𝑡 ∈ 𝑥 ∨ 𝑡 = 𝑦 ) ) )