Theorem tz9.1tco 36938

Description: Version of tz9.1 9697 derived from ax-tco 36927. (Contributed by Matthew House, 6-Apr-2026.)

Hypothesis

Ref	Expression
tz9.1tco.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tz9.1tco	⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem tz9.1tco

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tz9.1tco.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	tz9.1ctco 36937	. . 3 ⊢ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∈ V
3	2	isseti 3471	. 2 ⊢ ∃𝑥 𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
4		ssmin 4931	. . . 4 ⊢ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
5		sseq2 3962	. . . 4 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}))
6	4, 5	mpbiri 261	. . 3 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → 𝐴 ⊆ 𝑥)
7		treq 5224	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (Tr 𝑧 ↔ Tr 𝑦))
8	7	ralab2 3659	. . . . . 6 ⊢ (∀𝑧 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑧 ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦))
9		simpr 489	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦)
10	8, 9	mpgbir 1827	. . . . 5 ⊢ ∀𝑧 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑧
11		trint 5235	. . . . 5 ⊢ (∀𝑧 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑧 → Tr ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
12	10, 11	ax-mp 5	. . . 4 ⊢ Tr ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
13		treq 5224	. . . 4 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → (Tr 𝑥 ↔ Tr ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}))
14	12, 13	mpbiri 261	. . 3 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → Tr 𝑥)
15		eqimss 3994	. . . 4 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → 𝑥 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
16		ssintab 4929	. . . 4 ⊢ (𝑥 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
17	15, 16	sylib 221	. . 3 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
18	6, 14, 17	3jca 1144	. 2 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
19	3, 18	eximii 1865	1 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101  ∀wal 1566   = wceq 1568  ∃wex 1807   ∈ wcel 2141  {cab 2739  ∀wral 3077  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-nfc 2910  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-iin 4958  df-tr 5218

This theorem is referenced by: (None)