Skip to content
Axiomatic Wiki
Lemma·L07

Every natural number is transitive

Members of members of a natural are members: numbers nest cleanly.

in the sense of D024.
In words
For every n, if n is a natural number then n is a transitive set: a member of a member of n is itself a member of n.
Rests onF02F06F09F14A01A06
Never needed: F03 · F04 · F05 · F08 · F10 · F11 · F12 · F13 · A02 · A03 · A04 · A05 · A07 · A08 (computed from the citation graph, not asserted).

Proof

  1. 1
    Let (Separation); we show by induction.
  2. 2
    Base: is transitive vacuously: it has no members to check (D002, D024).
  3. 3
    Step: let be transitive; take (D006). If , then by transitivity of , and , so . If , then directly. Either way (case analysis), so is transitive and .
  4. 4
    T05 gives .
∎

Remarks

With the order , transitivity of each natural is exactly the transitivity of : from and , i.e. , transitivity of the set gives , i.e. . It also gives the useful reading of a natural as the set of all smaller naturals.
Wikipedia ↗

Used by

L15T12
Propose an edit1 published revision