Description: The Axiom of Transitive Containment of ZF set theory. It was derived as axtco above and is therefore redundant if we assume ax-ext , ax-rep and ax-inf2 , but we state it as a separate axiom here so that its uses can be identified more easily. It states that a transitive set y exists that contains a given set x . In particular, the transitive closure of x is a set, since it is a subset of y , see df-tc .
Traditionally, this statement is not counted as an axiom at all, but as a theorem from Replacement and Infinity. In fact, from the transitive closure of x we can construct the set of iterated unions of x (and vice versa), and Skolem took the existence of the latter set as a motivation for introducing the Axiom of Replacement. But Transitive Containment is strictly weaker than either of those axioms, so many authors identify it as its own axiom when investigating subsystems of ZF, such as Zermelo set theory or finitist set theory. We follow this separation in order to avoid nonessential usage of the stronger axioms.
There are two main versions of this axiom that appear in the literature: thestrong form |- E. y ( x e. y /\ Tr y ) , see axtco1 and axtco1g , and theweak form |- E. y ( x C_ y /\ Tr y ) , see axtco2 and axtco2g . The weak form follows directly from the strong form, see axtco2 . But the strong form only follows from the weak form if we allow el or one of its variants, see axtco1from2 . We take the strong form here as the axiom, since it is slightly shorter when expanded to primitive symbols. Yet the weak form turns out to be more suitable for axtcond for reasons of syntax. (Contributed by Matthew House, 6-Apr-2026)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-tco | |- E. y ( x e. y /\ A. z ( z e. y -> A. w ( w e. z -> w e. y ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | vy | |- y |
|
| 1 | vx | |- x |
|
| 2 | 1 | cv | |- x |
| 3 | 0 | cv | |- y |
| 4 | 2 3 | wcel | |- x e. y |
| 5 | vz | |- z |
|
| 6 | 5 | cv | |- z |
| 7 | 6 3 | wcel | |- z e. y |
| 8 | vw | |- w |
|
| 9 | 8 | cv | |- w |
| 10 | 9 6 | wcel | |- w e. z |
| 11 | 9 3 | wcel | |- w e. y |
| 12 | 10 11 | wi | |- ( w e. z -> w e. y ) |
| 13 | 12 8 | wal | |- A. w ( w e. z -> w e. y ) |
| 14 | 7 13 | wi | |- ( z e. y -> A. w ( w e. z -> w e. y ) ) |
| 15 | 14 5 | wal | |- A. z ( z e. y -> A. w ( w e. z -> w e. y ) ) |
| 16 | 4 15 | wa | |- ( x e. y /\ A. z ( z e. y -> A. w ( w e. z -> w e. y ) ) ) |
| 17 | 16 0 | wex | |- E. y ( x e. y /\ A. z ( z e. y -> A. w ( w e. z -> w e. y ) ) ) |