Theorem intss 3948

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 1561	. . . 4 ⊢ (∀y(y ∈ A → y ∈ B) → (∀y(y ∈ B → x ∈ y) → ∀y(y ∈ A → x ∈ y)))
3		vex 2863	. . . . 5 ⊢ x ∈ V
4	3	elint 3933	. . . 4 ⊢ (x ∈ ∩B ↔ ∀y(y ∈ B → x ∈ y))
5	3	elint 3933	. . . 4 ⊢ (x ∈ ∩A ↔ ∀y(y ∈ A → x ∈ y))
6	2, 4, 5	3imtr4g 261	. . 3 ⊢ (∀y(y ∈ A → y ∈ B) → (x ∈ ∩B → x ∈ ∩A))
7	6	alrimiv 1631	. 2 ⊢ (∀y(y ∈ A → y ∈ B) → ∀x(x ∈ ∩B → x ∈ ∩A))
8		dfss2 3263	. 2 ⊢ (A ⊆ B ↔ ∀y(y ∈ A → y ∈ B))
9		dfss2 3263	. 2 ⊢ (∩B ⊆ ∩A ↔ ∀x(x ∈ ∩B → x ∈ ∩A))
10	7, 8, 9	3imtr4i 257	1 ⊢ (A ⊆ B → ∩B ⊆ ∩A)

This proof depends on syntax axioms:   → wi 4  ∀wal 1540   ∈ wcel 1710   ⊆ wss 3258  ∩cint 3927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-int 3928

This theorem is used by:  uniintsn  3964