Theorem trin 5295

Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)

Assertion

Ref	Expression
trin	⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))

Proof of Theorem trin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3992	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2		trss 5294	. . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
3		trss 5294	. . . . . 6 ⊢ (Tr 𝐵 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ 𝐵))
4	2, 3	im2anan9 619	. . . . 5 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)))
5	1, 4	biimtrid 242	. . . 4 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → (𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵)))
6		ssin 4260	. . . 4 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵))
7	5, 6	imbitrdi 251	. . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ⊆ (𝐴 ∩ 𝐵)))
8	7	ralrimiv 3151	. 2 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵))
9		dftr3 5289	. 2 ⊢ (Tr (𝐴 ∩ 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐵)𝑥 ⊆ (𝐴 ∩ 𝐵))
10	8, 9	sylibr 234	1 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976  Tr wtr 5283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-v 3490  df-in 3983  df-ss 3993  df-uni 4932  df-tr 5284

This theorem is referenced by:  ordin  6425  tcmin  9810  ingru  10884  gruina  10887  dfon2lem4  35750