Theorem tz9.1ctco 37088

Description: Version of tz9.1c 9712 derived from ax-tco 37078. (Contributed by Matthew House, 6-Apr-2026.)

Hypothesis

Ref	Expression
tz9.1ctco.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tz9.1ctco	⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

Distinct variable group:   𝑥,𝐴

Proof of Theorem tz9.1ctco

Step	Hyp	Ref	Expression
1		tz9.1ctco.1	. . 3 ⊢ 𝐴 ∈ V
2		axtco2g 37083	. . 3 ⊢ (𝐴 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
3	1, 2	ax-mp 5	. 2 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥)
4		intexab 5314	. 2 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
5	3, 4	mpbi 233	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  {cab 2740  Vcvv 3453   ⊆ wss 3902  ∩ cint 4910  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-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-tco 37078

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-in 3909  df-ss 3919  df-nul 4283  df-uni 4871  df-int 4911  df-tr 5217

This theorem is used by:  tz9.1tco  37089