Theorem trint 4116

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 4105	. . . . . 6 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
2	1	ralbii 2483	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
3	2	biimpi 120	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
4		df-ral 2460	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
5	4	ralbii 2483	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
6		ralcom4 2759	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
7	5, 6	bitri 184	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
8	3, 7	sylib 122	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
9		ralim 2536	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
10	9	alimi 1455	. . 3 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥) → ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
11	8, 10	syl 14	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
12		dftr3 4105	. . 3 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑦 ∈ ∩ 𝐴𝑦 ⊆ ∩ 𝐴)
13		df-ral 2460	. . . 4 ⊢ (∀𝑦 ∈ ∩ 𝐴𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ ∩ 𝐴 → 𝑦 ⊆ ∩ 𝐴))
14		vex 2740	. . . . . . 7 ⊢ 𝑦 ∈ V
15	14	elint2 3851	. . . . . 6 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
16		ssint 3860	. . . . . 6 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
17	15, 16	imbi12i 239	. . . . 5 ⊢ ((𝑦 ∈ ∩ 𝐴 → 𝑦 ⊆ ∩ 𝐴) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
18	17	albii 1470	. . . 4 ⊢ (∀𝑦(𝑦 ∈ ∩ 𝐴 → 𝑦 ⊆ ∩ 𝐴) ↔ ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
19	13, 18	bitri 184	. . 3 ⊢ (∀𝑦 ∈ ∩ 𝐴𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
20	12, 19	bitri 184	. 2 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
21	11, 20	sylibr 134	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Syntax hints:   → wi 4  ∀wal 1351   ∈ wcel 2148  ∀wral 2455   ⊆ wss 3129  ∩ cint 3844  Tr wtr 4101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-uni 3810  df-int 3845  df-tr 4102

This theorem is referenced by:  onintonm  4516