Theorem axtco1g 37082

Description: Strong form of the Axiom of Transitive Containment using class variables and abbreviations. See ax-tco 37078 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 2850	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
2		dftr3 5221	. . . . . 6 ⊢ (Tr 𝑥 ↔ ∀𝑧 ∈ 𝑥 𝑧 ⊆ 𝑥)
3		df-ss 3919	. . . . . . 7 ⊢ (𝑧 ⊆ 𝑥 ↔ ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))
4	3	ralbii 3110	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 𝑧 ⊆ 𝑥 ↔ ∀𝑧 ∈ 𝑥 ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))
5		df-ral 3079	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) ↔ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)))
6	2, 4, 5	3bitrri 301	. . . . 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 644	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))) ↔ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)))
9	8	exbidv 1954	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥)))
10		axtco1 37079	. 2 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥)))
11	9, 10	vtoclg 3520	1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ∈ 𝑥 ∧ Tr 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  Tr wtr 5216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-tco 37078

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3455  df-ss 3919  df-uni 4871  df-tr 5217

This theorem is used by:  axtco2g  37083