Theorem axtco2g 37016

Description: Weak form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 37011 for more information. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
axtco2g	⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem axtco2g

Step	Hyp	Ref	Expression
1		axtco1g 37015	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))
2		trss 5227	. . . 4 ⊢ (Tr 𝑥 → (𝐴 ∈ 𝑥 → 𝐴 ⊆ 𝑥))
3	2	imdistanri 579	. . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
4	3	eximi 1864	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
5	1, 4	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∃wex 1808   ∈ wcel 2142   ⊆ wss 3904  Tr wtr 5217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-tco 37011

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3456  df-ss 3921  df-uni 4872  df-tr 5218

This theorem is used by:  tz9.1ctco  37021