Theorem tz9.1ctco 37448

Description: Version of tz9.1c 9758 derived from ax-tco 37438. (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 37443	. . 3 ⊢ (𝐴 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
3	1, 2	ax-mp 5	. 2 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥)
4		intexab 5311	. 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 2147  {cab 2741  Vcvv 3454   ⊆ wss 3900  ∩ cint 4910  Tr wtr 5215

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-tco 37438

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-nul 4282  df-uni 4871  df-int 4911  df-tr 5216

This theorem is used by:  tz9.1tco  37449