Theorem tz9.1tco 37313

Description: Version of tz9.1 9730 derived from ax-tco 37302. (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 37312	. . 3 ⊢ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ∈ V
3	2	isseti 3468	. 2 ⊢ ∃𝑥 𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
4		ssmin 4926	. . . 4 ⊢ 𝐴 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}
5		sseq2 3934	. . . 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 5218	. . . . . . 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 490	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦)
10	8, 9	mpgbir 1831	. . . . 5 ⊢ ∀𝑧 ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}Tr 𝑧
11		trint 5229	. . . . 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 5218	. . . 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 3966	. . . 4 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → 𝑥 ⊆ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)})
16		ssintab 4924	. . . 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 1146	. 2 ⊢ (𝑥 = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → (𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
19	3, 18	eximii 1869	1 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ∩ cint 4906  Tr wtr 5211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-tco 37302

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-nul 4279  df-uni 4867  df-int 4907  df-iin 4953  df-tr 5212

This theorem is used by: (None)