Theorem trint 5224

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 5223	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝑥 ∈ 𝐴 𝑥)
2		intiin 5016	. . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥
3		treq 5213	. . 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 236	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559  ∀wral 3075  ∩ cint 4904  ∩ ciin 4949  Tr wtr 5206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-v 3455  df-ss 3921  df-uni 4865  df-int 4905  df-iin 4951  df-tr 5207

This theorem is referenced by:  tctr  9690  intwun  10690  intgru  10769  tz9.1regs  35394  dfon2lem8  36102  tz9.1tco  36807  dfttc3gw  36847