Theorem trint 5301

Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by BJ, 3-Oct-2022.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem trint

Step	Hyp	Ref	Expression
1		triin 5300	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 5082	. . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3		treq 5291	. . 3 ⊢ (∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 → (Tr ∩ 𝐴 ↔ Tr ∩ 𝑥 ∈ 𝐴 𝑥))
4	2, 3	ax-mp 5	. 2 ⊢ (Tr ∩ 𝐴 ↔ Tr ∩ 𝑥 ∈ 𝐴 𝑥)
5	1, 4	sylibr 234	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  ∀wral 3067  ∩ cint 4970  ∩ ciin 5016  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-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-v 3490  df-ss 3993  df-uni 4932  df-int 4971  df-iin 5018  df-tr 5284

This theorem is referenced by:  tctr  9809  intwun  10804  intgru  10883  dfon2lem8  35754