Theorem axtco2g 36842

Description: Weak form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36837 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 36841	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))
2		trss 5219	. . . 4 ⊢ (Tr 𝑥 → (𝐴 ∈ 𝑥 → 𝐴 ⊆ 𝑥))
3	2	imdistanri 577	. . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
4	3	eximi 1857	. 2 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥) → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
5	1, 4	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1801   ∈ wcel 2144   ⊆ wss 3906  Tr wtr 5209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-tco 36837

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-v 3458  df-ss 3923  df-uni 4868  df-tr 5210

This theorem is referenced by:  tz9.1ctco  36847