Theorem axtco1g 36664

Description: Strong form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36660 for more information. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
axtco1g	⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem axtco1g

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2825	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2		dftr3 5198	. . . . . 6 ⊢ (Tr 𝑥 ↔ ∀𝑧 ∈ 𝑥 𝑧 ⊆ 𝑥)
3		df-ss 3907	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑥 ↔ ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))
4	3	ralbii 3084	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 𝑧 ⊆ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))
5		df-ral 3053	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)))
6	2, 4, 5	3bitrri 298	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)) ↔ Tr 𝑥)
7	6	a1i 11	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)) ↔ Tr 𝑥))
8	1, 7	anbi12d 633	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))) ↔ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)))
9	8	exbidv 1923	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥)))
10		axtco1 36661	. 2 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)))
11	9, 10	vtoclg 3500	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  Tr wtr 5193

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 2709  ax-tco 36660

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3432  df-ss 3907  df-uni 4852  df-tr 5194

This theorem is referenced by:  axtco2g  36665