Theorem intss 3627

Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)

Assertion

Ref	Expression
intss	⊢ (A ⊆ B → ∩ B ⊆ ∩ A)

Proof of Theorem intss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imim1 70	. . . . 5 ⊢ ((y ∈ A → y ∈ B) → ((y ∈ B → x ∈ y) → (y ∈ A → x ∈ y)))
2	1	al2imi 1344	. . . 4 ⊢ (∀y(y ∈ A → y ∈ B) → (∀y(y ∈ B → x ∈ y) → ∀y(y ∈ A → x ∈ y)))
3		vex 2554	. . . . 5 ⊢ x ∈ V
4	3	elint 3612	. . . 4 ⊢ (x ∈ ∩ B ↔ ∀y(y ∈ B → x ∈ y))
5	3	elint 3612	. . . 4 ⊢ (x ∈ ∩ A ↔ ∀y(y ∈ A → x ∈ y))
6	2, 4, 5	3imtr4g 194	. . 3 ⊢ (∀y(y ∈ A → y ∈ B) → (x ∈ ∩ B → x ∈ ∩ A))
7	6	alrimiv 1751	. 2 ⊢ (∀y(y ∈ A → y ∈ B) → ∀x(x ∈ ∩ B → x ∈ ∩ A))
8		dfss2 2928	. 2 ⊢ (A ⊆ B ↔ ∀y(y ∈ A → y ∈ B))
9		dfss2 2928	. 2 ⊢ (∩ B ⊆ ∩ A ↔ ∀x(x ∈ ∩ B → x ∈ ∩ A))
10	7, 8, 9	3imtr4i 190	1 ⊢ (A ⊆ B → ∩ B ⊆ ∩ A)

Syntax hints:   → wi 4  ∀wal 1240   ∈ wcel 1390   ⊆ wss 2911  ∩ cint 3606

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-int 3607

This theorem is referenced by: (None)