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Theorem inss1 3154
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 3123 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21simplbi 259 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
32ssriv 2946 1 (𝐴𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1393  cin 2913  wss 2914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-in 2921  df-ss 2928
This theorem is referenced by:  inss2  3155  ssinss1  3162  unabs  3164  nssinpss  3166  inssddif  3175  inv1  3250  disjdif  3293  inundifss  3298  relin1  4433  resss  4613  resmpt3  4635  cnvcnvss  4753  funin  4948  funimass2  4955  fnresin1  4991  fnres  4993  fresin  5046  ssimaex  5212  fneqeql2  5254  isoini2  5436  ofrfval  5698  fnofval  5699  ofrval  5700  off  5702  ofres  5703  ofco  5707  smores  5885  smores2  5887  tfrlem5  5908  peano5nnnn  6938  peano5nni  7884  rexanuz  9456
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