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Theorem ssin 3994
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 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))

Proof of Theorem ssin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3958 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
21imbi2i 327 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
32albii 1914 . . 3 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
4 jcab 513 . . . 4 ((𝑥𝐴 → (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
54albii 1914 . . 3 (∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)) ↔ ∀𝑥((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
6 19.26 1968 . . 3 (∀𝑥((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)))
73, 5, 63bitrri 289 . 2 ((∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
8 dfss2 3749 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
9 dfss2 3749 . . 3 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
108, 9anbi12i 620 . 2 ((𝐴𝐵𝐴𝐶) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)))
11 dfss2 3749 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
127, 10, 113bitr4i 294 1 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wcel 2155  cin 3731  wss 3732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3739  df-ss 3746
This theorem is referenced by:  ssini  3995  ssind  3996  uneqin  4043  disjpss  4189  trin  4921  pwin  5179  fin  6267  wfrlem4  7621  wfrlem4OLD  7622  epfrs  8822  tcmin  8832  resscntz  18027  subgdmdprd  18700  tgval  21039  eltg3i  21045  innei  21209  cnprest2  21374  subislly  21564  lly1stc  21579  xkohaus  21736  xkoinjcn  21770  opnfbas  21925  supfil  21978  rnelfm  22036  tsmsres  22226  restmetu  22654  chabs2  28832  cmbr4i  28916  pjin3i  29509  mdbr2  29611  dmdbr2  29618  dmdbr5  29623  mdslle1i  29632  mdslle2i  29633  mdslj1i  29634  mdslj2i  29635  mdsl2i  29637  mdslmd1lem1  29640  mdslmd1lem2  29641  mdslmd1i  29644  mdslmd3i  29647  hatomistici  29677  chrelat2i  29680  cvexchlem  29683  mdsymlem1  29718  mdsymlem3  29720  mdsymlem6  29723  dmdbr5ati  29737  pnfneige0  30444  ballotlem2  30998  iccllysconn  31680  frrlem4  32227  heibor1lem  34030  dochexmidlem1  37416  superficl  38547  k0004lem1  39119
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