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Theorem inss1 3392
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 3355 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21simplbi 274 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
32ssriv 3196 1 (𝐴𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2175  cin 3164  wss 3165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186
This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178
This theorem is referenced by:  inss2  3393  ssinss1  3401  unabs  3403  inssddif  3413  inv1  3496  disjdif  3532  inundifss  3537  relin1  4792  resss  4982  resmpt3  5007  cnvcnvss  5136  funin  5344  funimass2  5351  fnresin1  5389  fnres  5391  fresin  5453  ssimaex  5639  fneqeql2  5688  isoini2  5887  ofrfval  6166  ofvalg  6167  ofrval  6168  off  6170  ofres  6172  ofco  6176  smores  6377  smores2  6379  tfrlem5  6399  pmresg  6762  unfiin  7022  infidc  7035  sbthlem7  7064  peano5nnnn  8004  peano5nni  9038  rexanuz  11270  nninfdclemcl  12790  nninfdclemp1  12792  fvsetsid  12837  tgvalex  13066  tgval2  14494  eltg3  14500  tgcl  14507  tgdom  14515  tgidm  14517  epttop  14533  ntropn  14560  ntrin  14567  cnptopresti  14681  cnptoprest  14682  txcnmpt  14716  xmetres  14825  metres  14826  blin2  14875  metrest  14949  tgioo  14997  limcresi  15109  2sqlem8  15571  bj-charfun  15705  bj-charfundc  15706  bj-charfundcALT  15707
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