Theorem trint 4768

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.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem trint

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr3 4756	. . . . 5 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
2	1	ralbii 2980	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
3		df-ral 2917	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
4	3	ralbii 2980	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
5		ralcom4 3224	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
6	4, 5	bitri 264	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
7	2, 6	sylbb 209	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
8		ralim 2948	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
9	7, 8	sylg 1750	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
10		dftr3 4756	. . 3 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑦 ∈ ∩ 𝐴𝑦 ⊆ ∩ 𝐴)
11		df-ral 2917	. . . 4 ⊢ (∀𝑦 ∈ ∩ 𝐴𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ ∩ 𝐴 → 𝑦 ⊆ ∩ 𝐴))
12		vex 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
13	12	elint2 4482	. . . . . 6 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
14		ssint 4493	. . . . . 6 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
15	13, 14	imbi12i 340	. . . . 5 ⊢ ((𝑦 ∈ ∩ 𝐴 → 𝑦 ⊆ ∩ 𝐴) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
16	15	albii 1747	. . . 4 ⊢ (∀𝑦(𝑦 ∈ ∩ 𝐴 → 𝑦 ⊆ ∩ 𝐴) ↔ ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
17	11, 16	bitri 264	. . 3 ⊢ (∀𝑦 ∈ ∩ 𝐴𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
18	10, 17	bitri 264	. 2 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
19	9, 18	sylibr 224	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Syntax hints:   → wi 4  ∀wal 1481   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  ∩ cint 4475  Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-int 4476  df-tr 4753

This theorem is referenced by:  tctr  8616  intwun  9557  intgru  9636  dfon2lem8  31695