Theorem tz9.1ctco 36937

Description: Version of tz9.1c 9698 derived from ax-tco 36927. (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 36932	. . 3 ⊢ (𝐴 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
3	1, 2	ax-mp 5	. 2 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥)
4		intexab 5316	. 2 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
5	3, 4	mpbi 233	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

Syntax hints:   ∧ wa 400  ∃wex 1807   ∈ wcel 2141  {cab 2739  Vcvv 3453   ⊆ wss 3904  ∩ cint 4911  Tr wtr 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-tco 36927

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286  df-uni 4872  df-int 4912  df-tr 5218

This theorem is referenced by:  tz9.1tco  36938