ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inss1 GIF version

Theorem inss1 3266
Description: The intersection of two classes is a subset of one of them. Part of Exercise 12 of [TakeutiZaring] p. 18. (Contributed by NM, 27-Apr-1994.)
Assertion
Ref Expression
inss1 (𝐴𝐵) ⊆ 𝐴

Proof of Theorem inss1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elin 3229 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21simplbi 272 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
32ssriv 3071 1 (𝐴𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1465  cin 3040  wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054
This theorem is referenced by:  inss2  3267  ssinss1  3275  unabs  3277  inssddif  3287  inv1  3369  disjdif  3405  inundifss  3410  relin1  4627  resss  4813  resmpt3  4838  cnvcnvss  4963  funin  5164  funimass2  5171  fnresin1  5207  fnres  5209  fresin  5271  ssimaex  5450  fneqeql2  5497  isoini2  5688  ofrfval  5958  ofvalg  5959  ofrval  5960  off  5962  ofres  5964  ofco  5968  smores  6157  smores2  6159  tfrlem5  6179  pmresg  6538  unfiin  6782  sbthlem7  6819  peano5nnnn  7668  peano5nni  8687  rexanuz  10715  fvsetsid  11904  tgvalex  12130  tgval2  12131  eltg3  12137  tgcl  12144  tgdom  12152  tgidm  12154  epttop  12170  ntropn  12197  ntrin  12204  cnptopresti  12318  cnptoprest  12319  txcnmpt  12353  xmetres  12462  metres  12463  blin2  12512  metrest  12586  tgioo  12626  limcresi  12715
  Copyright terms: Public domain W3C validator