Theorem trint 5276

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 5275	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 5058	. . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3		treq 5266	. . 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 1539  ∀wral 3060  ∩ cint 4945  ∩ ciin 4991  Tr wtr 5258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-v 3481  df-ss 3967  df-uni 4907  df-int 4946  df-iin 4993  df-tr 5259

This theorem is referenced by:  tctr  9781  intwun  10776  intgru  10855  dfon2lem8  35792