Theorem tz9.1ctco 36847

Description: Version of tz9.1c 9683 derived from ax-tco 36837. (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 36842	. . 3 ⊢ (𝐴 ∈ V → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥))
3	1, 2	ax-mp 5	. 2 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥)
4		intexab 5303	. 2 ⊢ (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
5	3, 4	mpbi 232	1 ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V

Syntax hints:   ∧ wa 399  ∃wex 1800   ∈ wcel 2143  {cab 2741  Vcvv 3455   ⊆ wss 3905  ∩ cint 4906  Tr wtr 5208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-tco 36837

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-in 3912  df-ss 3922  df-nul 4287  df-uni 4867  df-int 4907  df-tr 5209

This theorem is referenced by:  tz9.1tco  36848