Theorem intss1 3942

Description: An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)

Assertion

Ref	Expression
intss1	⊢ (A ∈ B → ∩B ⊆ A)

Proof of Theorem intss1

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

Step	Hyp	Ref	Expression
1		vex 2863	. . . 4 ⊢ x ∈ V
2	1	elint 3933	. . 3 ⊢ (x ∈ ∩B ↔ ∀y(y ∈ B → x ∈ y))
3		eleq1 2413	. . . . . 6 ⊢ (y = A → (y ∈ B ↔ A ∈ B))
4		eleq2 2414	. . . . . 6 ⊢ (y = A → (x ∈ y ↔ x ∈ A))
5	3, 4	imbi12d 311	. . . . 5 ⊢ (y = A → ((y ∈ B → x ∈ y) ↔ (A ∈ B → x ∈ A)))
6	5	spcgv 2940	. . . 4 ⊢ (A ∈ B → (∀y(y ∈ B → x ∈ y) → (A ∈ B → x ∈ A)))
7	6	pm2.43a 45	. . 3 ⊢ (A ∈ B → (∀y(y ∈ B → x ∈ y) → x ∈ A))
8	2, 7	syl5bi 208	. 2 ⊢ (A ∈ B → (x ∈ ∩B → x ∈ A))
9	8	ssrdv 3279	1 ⊢ (A ∈ B → ∩B ⊆ A)

Syntax hints:   → wi 4  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ⊆ wss 3258  ∩cint 3927

This theorem was proved from 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 theorem 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 referenced by:  intminss  3953  intmin3  3955  intab  3957  int0el  3958  peano5  4410  spfininduct  4541  clos1induct  5881