Theorem ssin 4161

Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem ssin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3899	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶))
2	1	imbi2i 335	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
3	2	albii 1823	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)))
4		jcab 517	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
5	4	albii 1823	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)) ↔ ∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
6		19.26 1874	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
7	3, 5, 6	3bitrri 297	. 2 ⊢ ((∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)))
8		dfss2 3903	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
9		dfss2 3903	. . 3 ⊢ (𝐴 ⊆ 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
10	8, 9	anbi12i 626	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶)))
11		dfss2 3903	. 2 ⊢ (𝐴 ⊆ (𝐵 ∩ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐵 ∩ 𝐶)))
12	7, 10, 11	3bitr4i 302	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   ∈ wcel 2108   ∩ cin 3882   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by:  ssini  4162  ssind  4163  uneqin  4209  disjpss  4391  trin  5197  pwin  5474  fin  6638  frrlem4  8076  frrlem13  8085  wfrlem4OLD  8114  epfrs  9420  tcmin  9430  resscntz  18853  subgdmdprd  19552  tgval  22013  eltg3i  22019  innei  22184  cnprest2  22349  subislly  22540  lly1stc  22555  xkohaus  22712  xkoinjcn  22746  opnfbas  22901  supfil  22954  rnelfm  23012  tsmsres  23203  restmetu  23632  chabs2  29780  cmbr4i  29864  pjin3i  30457  mdbr2  30559  dmdbr2  30566  dmdbr5  30571  mdslle1i  30580  mdslle2i  30581  mdslj1i  30582  mdslj2i  30583  mdsl2i  30585  mdslmd1lem1  30588  mdslmd1lem2  30589  mdslmd1i  30592  mdslmd3i  30595  hatomistici  30625  chrelat2i  30628  cvexchlem  30631  mdsymlem1  30666  mdsymlem3  30668  mdsymlem6  30671  dmdbr5ati  30685  pnfneige0  31803  ballotlem2  32355  iccllysconn  33112  heibor1lem  35894  dochexmidlem1  39401  superficl  41063  k0004lem1  41646  ismnushort  41808