Axiom ax-tco 37437

Description: The Axiom of Transitive Containment of ZF set theory. It was derived as axtco 37436 above and is therefore redundant if we assume ax-ext 2735, ax-rep 5235 and ax-inf2 9666, 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 9760.

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: the strong form ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ Tr 𝑦), see axtco1 37438 and axtco1g 37441, and the weak form ⊢ ∃𝑦(𝑥 ⊆ 𝑦 ∧ Tr 𝑦), see axtco2 37439 and axtco2g 37442. The weak form follows directly from the strong form, see axtco2 37439. But the strong form only follows from the weak form if we allow el 5410 or one of its variants, see axtco1from2 37440. 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 37443 for reasons of syntax. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ax-tco	⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Detailed syntax breakdown of Axiom ax-tco

Step	Hyp	Ref	Expression
1		vx	. . . 4 setvar 𝑥
2		vy	. . . 4 setvar 𝑦
3	1, 2	wel 2148	. . 3 wff 𝑥 ∈ 𝑦
4		vz	. . . . . 6 setvar 𝑧
5	4, 2	wel 2148	. . . . 5 wff 𝑧 ∈ 𝑦
6		vw	. . . . . . . 8 setvar 𝑤
7	6, 4	wel 2148	. . . . . . 7 wff 𝑤 ∈ 𝑧
8	6, 2	wel 2148	. . . . . . 7 wff 𝑤 ∈ 𝑦
9	7, 8	wi 4	. . . . . 6 wff (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)
10	9, 6	wal 1568	. . . . 5 wff ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)
11	5, 10	wi 4	. . . 4 wff (𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))
12	11, 4	wal 1568	. . 3 wff ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))
13	3, 12	wa 401	. 2 wff (𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))
14	13, 2	wex 1812	1 wff ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦)))

This axiom is used by:  axtco1  37438  axuntco  37444  axnulregtco  37445  elALTtco  37446