Theorem ssin 1659

Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26.

Assertion

Ref	Expression
ssin	⊢ ((A ⊆ B ∧ A ⊆ C) ↔ A ⊆ (B ∩ C))

Proof of Theorem ssin

Step	Hyp	Ref	Expression
1		ineq12 1640	. . . . 5 ⊢ (((A ∩ B) = A ∧ (A ∩ C) = A) → ((A ∩ B) ∩ (A ∩ C)) = (A ∩ A))
2		inindi 1654	. . . . 5 ⊢ (A ∩ (B ∩ C)) = ((A ∩ B) ∩ (A ∩ C))
3	1, 2	syl5eq 1136	. . . 4 ⊢ (((A ∩ B) = A ∧ (A ∩ C) = A) → (A ∩ (B ∩ C)) = (A ∩ A))
4		inidm 1649	. . . 4 ⊢ (A ∩ A) = A
5	3, 4	syl6eq 1140	. . 3 ⊢ (((A ∩ B) = A ∧ (A ∩ C) = A) → (A ∩ (B ∩ C)) = A)
6		df-ss 1492	. . . 4 ⊢ (A ⊆ B ↔ (A ∩ B) = A)
7		df-ss 1492	. . . 4 ⊢ (A ⊆ C ↔ (A ∩ C) = A)
8	6, 7	anbi12i 369	. . 3 ⊢ ((A ⊆ B ∧ A ⊆ C) ↔ ((A ∩ B) = A ∧ (A ∩ C) = A))
9		df-ss 1492	. . 3 ⊢ (A ⊆ (B ∩ C) ↔ (A ∩ (B ∩ C)) = A)
10	5, 8, 9	3imtr4 192	. 2 ⊢ ((A ⊆ B ∧ A ⊆ C) → A ⊆ (B ∩ C))
11		inss1 1657	. . . 4 ⊢ (B ∩ C) ⊆ B
12		sstr2 1510	. . . 4 ⊢ (A ⊆ (B ∩ C) → ((B ∩ C) ⊆ B → A ⊆ B))
13	11, 12	mpi 44	. . 3 ⊢ (A ⊆ (B ∩ C) → A ⊆ B)
14		inss2 1658	. . . 4 ⊢ (B ∩ C) ⊆ C
15		sstr2 1510	. . . 4 ⊢ (A ⊆ (B ∩ C) → ((B ∩ C) ⊆ C → A ⊆ C))
16	14, 15	mpi 44	. . 3 ⊢ (A ⊆ (B ∩ C) → A ⊆ C)
17	13, 16	jca 236	. 2 ⊢ (A ⊆ (B ∩ C) → (A ⊆ B ∧ A ⊆ C))
18	10, 17	impbi 139	1 ⊢ ((A ⊆ B ∧ A ⊆ C) ↔ A ⊆ (B ∩ C))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∩ cin 1486   ⊆ wss 1487

This theorem is referenced by:  ssini 1660  nssinpss 1665  disjpss 1738  pwin 1915  trin 2051  fin 2770  zfregs 3491  chabs2t 5433  cmbr4 5510  pjin3 5648  mdbr2 5728  dmdbr2 5733  hatomistic 5755  chrelat2 5758  cvexchlem 5759  mdsymlem1 5776  mdsymlem3 5778  mdsymlem5 5780  mdsymlem6 5781

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492

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