Theorem tz9.1tco 36791

Description: Version of tz9.1 9674 derived from ax-tco 36780. (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 36790	. . 3 ⊢ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∈ V
3	2	isseti 3466	. 2 ⊢ ∃𝑥 𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
4		ssmin 4919	. . . 4 ⊢ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
5		sseq2 3957	. . . 4 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}))
6	4, 5	mpbiri 260	. . 3 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → 𝐴 ⊆ 𝑥)
7		treq 5208	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (Tr 𝑧 ↔ Tr 𝑦))
8	7	ralab2 3654	. . . . . 6 ⊢ (∀𝑧 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑧 ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦))
9		simpr 487	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦)
10	8, 9	mpgbir 1813	. . . . 5 ⊢ ∀𝑧 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑧
11		trint 5219	. . . . 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 5208	. . . 4 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → (Tr 𝑥 ↔ Tr ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}))
14	12, 13	mpbiri 260	. . 3 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → Tr 𝑥)
15		eqimss 3989	. . . 4 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → 𝑥 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
16		ssintab 4917	. . . 4 ⊢ (𝑥 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
17	15, 16	sylib 220	. . 3 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
18	6, 14, 17	3jca 1137	. 2 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
19	3, 18	eximii 1851	1 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095  ∀wal 1552   = wceq 1554  ∃wex 1793   ∈ wcel 2136  {cab 2734  ∀wral 3070  Vcvv 3448   ⊆ wss 3899  ∩ cint 4899  Tr wtr 5201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-tco 36780

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4281  df-uni 4860  df-int 4900  df-iin 4946  df-tr 5202

This theorem is referenced by: (None)