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Theorem ssin 3978
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 3939 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
21imbi2i 325 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
32albii 1896 . . 3 (∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)))
4 jcab 943 . . . 4 ((𝑥𝐴 → (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
54albii 1896 . . 3 (∀𝑥(𝑥𝐴 → (𝑥𝐵𝑥𝐶)) ↔ ∀𝑥((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)))
6 19.26 1947 . . 3 (∀𝑥((𝑥𝐴𝑥𝐵) ∧ (𝑥𝐴𝑥𝐶)) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)))
73, 5, 63bitrri 287 . 2 ((∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
8 dfss2 3732 . . 3 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
9 dfss2 3732 . . 3 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
108, 9anbi12i 735 . 2 ((𝐴𝐵𝐴𝐶) ↔ (∀𝑥(𝑥𝐴𝑥𝐵) ∧ ∀𝑥(𝑥𝐴𝑥𝐶)))
11 dfss2 3732 . 2 (𝐴 ⊆ (𝐵𝐶) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐵𝐶)))
127, 10, 113bitr4i 292 1 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wcel 2139  cin 3714  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729
This theorem is referenced by:  ssini  3979  ssind  3980  uneqin  4021  disjpss  4172  trin  4915  pwin  5168  fin  6246  wfrlem4  7588  epfrs  8782  tcmin  8792  resscntz  17984  subgdmdprd  18653  tgval  20981  eltg3i  20987  innei  21151  cnprest2  21316  subislly  21506  lly1stc  21521  xkohaus  21678  xkoinjcn  21712  opnfbas  21867  supfil  21920  rnelfm  21978  tsmsres  22168  restmetu  22596  chabs2  28706  cmbr4i  28790  pjin3i  29383  mdbr2  29485  dmdbr2  29492  dmdbr5  29497  mdslle1i  29506  mdslle2i  29507  mdslj1i  29508  mdslj2i  29509  mdsl2i  29511  mdslmd1lem1  29514  mdslmd1lem2  29515  mdslmd1i  29518  mdslmd3i  29521  hatomistici  29551  chrelat2i  29554  cvexchlem  29557  mdsymlem1  29592  mdsymlem3  29594  mdsymlem6  29597  dmdbr5ati  29611  pnfneige0  30327  ballotlem2  30880  iccllysconn  31560  frrlem4  32110  heibor1lem  33939  dochexmidlem1  37269  superficl  38392  k0004lem1  38965
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