Ideas from 'Elements of Set Theory' by Herbert B. Enderton [1977], by Theme Structure
[found in 'Elements of Set Theory' by Enderton,Herbert B. [Posts + Telecoms 2006,7115145504]].
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
13206

A 'linear or total ordering' must be transitive and satisfy trichotomy

13201

∈ says the whole set is in the other; ⊆ says the members of the subset are in the other

13204

The 'ordered pair' <x,y> is defined to be {{x}, {x,y}}

4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
13200

Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ

13199

The empty set may look pointless, but many sets can be constructed from it

4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Set
13203

The singleton is defined using the pairing axiom (as {x,x})

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
13202

Fraenkel added Replacement, to give a theory of ordinal numbers

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13205

We can only define functions if Choice tells us which items are involved
