Theorem axtco1g 36717

Description: Strong form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 36713 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 2829	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2		dftr3 5186	. . . . . 6 ⊢ (Tr 𝑥 ↔ ∀𝑧 ∈ 𝑥 𝑧 ⊆ 𝑥)
3		df-ss 3901	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑥 ↔ ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))
4	3	ralbii 3087	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 𝑧 ⊆ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))
5		df-ral 3056	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)))
6	2, 4, 5	3bitrri 300	. . . . 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 639	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))) ↔ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)))
9	8	exbidv 1929	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥)))
10		axtco1 36714	. 2 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)))
11	9, 10	vtoclg 3501	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∀wral 3055   ⊆ wss 3884  Tr wtr 5181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-tco 36713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-v 3435  df-ss 3901  df-uni 4841  df-tr 5182

This theorem is referenced by:  axtco2g  36718