Theorem trintOLD 4960

Description: Obsolete proof of trintOLD 4960 as of 3-Oct-2022. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem trintOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr3 4947	. . . . 5 ⊢ (Tr 𝑥 ↔ ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
2	1	ralbii 3159	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥)
3		df-ral 3092	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
4	3	ralbii 3159	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
5		ralcom4 3410	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
6	4, 5	bitri 267	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
7	2, 6	sylbb 211	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥))
8		ralim 3127	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝑥 → 𝑦 ⊆ 𝑥) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
9	7, 8	sylg 1918	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
10		dftr3 4947	. . 3 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑦 ∈ ∩ 𝐴𝑦 ⊆ ∩ 𝐴)
11		df-ral 3092	. . . 4 ⊢ (∀𝑦 ∈ ∩ 𝐴𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑦(𝑦 ∈ ∩ 𝐴 → 𝑦 ⊆ ∩ 𝐴))
12		vex 3386	. . . . . . 7 ⊢ 𝑦 ∈ V
13	12	elint2 4672	. . . . . 6 ⊢ (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
14		ssint 4681	. . . . . 6 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
15	13, 14	imbi12i 342	. . . . 5 ⊢ ((𝑦 ∈ ∩ 𝐴 → 𝑦 ⊆ ∩ 𝐴) ↔ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
16	15	albii 1915	. . . 4 ⊢ (∀𝑦(𝑦 ∈ ∩ 𝐴 → 𝑦 ⊆ ∩ 𝐴) ↔ ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
17	11, 16	bitri 267	. . 3 ⊢ (∀𝑦 ∈ ∩ 𝐴𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
18	10, 17	bitri 267	. 2 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑦(∀𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
19	9, 18	sylibr 226	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Syntax hints:   → wi 4  ∀wal 1651   ∈ wcel 2157  ∀wral 3087   ⊆ wss 3767  ∩ cint 4665  Tr wtr 4943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-v 3385  df-in 3774  df-ss 3781  df-uni 4627  df-int 4666  df-tr 4944

This theorem is referenced by: (None)