Language defined in this other blog post.
WORK IN PROGRESS, SUBJECT TO CHANGE
Axioms for the ∈ relation:[]
Axiom of extensionality for non-link objects (changed to not imply that all members of are equal): ∀x∀y((x∈)(y∈)((∀z((z∈x)(z∈y)))x=y))
Axiom of regularity: ∀(∃(∈)∃∈(¬∃(∈∈)))
Axiom of pairing: ∀∀∃z∀y(y∈z(y=∨y=))
Axiom of union: ∀∃∀(∈∃(∈∧∈))
Axioms for , , , , and relations (originally introduced by P進大好きbot in User blog:P進大好きbot/Rayo name of 65536#Wedge):[]
Axiom of Wedging: ∀x∀y∀z((x,y,z)(yz∃((y∈z∈)∈x))))
Axiom of Uniquely Wedging: ∀x∀y∀z((x,y,z)((x,y,z)(¬∃((z(x,y,))))))
Axiom of Restriction Of Uniquely Wedging: ∀x∀y∀z((x,y,z)∃(∈y(x,,z)))
Axiom of Restriction Of Having Surjective Wedge Relation: ∀x∀y∀z((x,y,z)¬∃(∈z¬(x,y,)))
Axiom of Surjection: ∀x∀y((xy)∃z((z,x,y)))
Axiom for the relation:[]
Axiom of Equal Cardinality: ∀x∀y((xy)(((xy)(yx))x=y))
Axiom for the relation:[]
Axiom of unordered pair: ∀x((x)∃y ∈x(∃z ∈x(yz(∃ ∈x(yz)))))
Axiom for the relation:[]
Axiom of one element from each set in unordered pair: ∀x∀y∀z((x,y,z)(x∃∈x(∈y∃∈x(∈y))))
Axiom for the relation:[]
Axiom of containing only unordered pairs: ∀x((x)∀y∈x(y))
Axiom for the relation:[]
Axiom of double containing all elements: ∀x∀y(xy∀z∈x(∃∈y(z∈)))
Axiom for the relation:[]
Axiom of map: ∀x∀y∀z((x,y,z)(xyyzx(∀∈x((,y,z)))yxzx))
Axioms for the relation:[]
Axiom Of Subsets: ∀x∀y(yx∀z(z∈yz∈x))
Axiom of the Power Set: ∀x∃y∀z(zxz∈y)
Axiom for the function:[]
Axioms of Union: ∀x∀y∀z((z∈(xy))((z∈x)(z∈y))
Axiom for the function:[]
Axiom of Intersection: ∀x∀y∀z((z∈(xy))((z∈x)(z∈y))
Axiom for the S function:[]
Axiom of Successor: ∀x∀y((y∈Sx)((x=y)(x∈y)))
Axioms for the 🔗 relation:[]
Axiom of link extensionality: ∀x∈∀y∈((∀z∈((x🔗z)(y🔗z)))x=y)
Axiom of Commutativity: ∀x∈∀y∈((x🔗y)(y🔗x))
Axiom of Separate Links: ∀x(∃y∈(∀z∈x((((¬(x=))(¬(y=z)))¬(y🔗z))))
Axiom of Link Existence to Alternate Set of Links:
🔗🔗
🔗
Axiom of emptiness: ∀x∈∀y(¬(y∈x))
Axiom of if sets can link: ∀x((¬(x∈))∀y(¬(x🔗y)))
Other axiom:[]
Axiom of infinity which contains no link: ∃(∃y∈(¬∃z(z∈y))∀∈(S∈)¬∃y∈(y∈))