Theorem axtco2g 36659

Description: Weak form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36654 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 36658	. 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))
2		trss 5202	. . . 4 ⊢ (Tr 𝑥 → (𝐴 ∈ 𝑥 → 𝐴 ⊆ 𝑥))
3	2	imdistanri 569	. . 3 ⊢ ((𝐴 ∈ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
4	3	eximi 1837	. 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 395  ∃wex 1781   ∈ wcel 2114   ⊆ wss 3889  Tr wtr 5192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-tco 36654

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3431  df-ss 3906  df-uni 4851  df-tr 5193

This theorem is referenced by:  tz9.1ctco  36664