Theorem triin 5228

Description: An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.)

Assertion

Ref	Expression
triin	⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵)

Proof of Theorem triin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliin 4955	. . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
2	1	elv 3455	. . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3		r19.26 3122	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
4		trss 5221	. . . . . . . 8 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ⊆ 𝐵))
5	4	imp 412	. . . . . . 7 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐵)
6	5	ralimi 3099	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵)
7	3, 6	sylbir 238	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵)
8		ssiin 5013	. . . . 5 ⊢ (𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵)
9	7, 8	sylibr 237	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
10	2, 9	sylan2b 606	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
11	10	ralrimiva 3154	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
12		dftr3 5216	. 2 ⊢ (Tr ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
13	11, 12	sylibr 237	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ∩ ciin 4951  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-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-v 3452  df-ss 3896  df-uni 4867  df-iin 4953  df-tr 5212

This theorem is used by:  trint  5229