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Theorem inss1 3264
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 3227 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
21simplbi 270 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥𝐴)
32ssriv 3069 1 (𝐴𝐵) ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 1463  cin 3038  wss 3039
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052
This theorem is referenced by:  inss2  3265  ssinss1  3273  unabs  3275  inssddif  3285  inv1  3367  disjdif  3403  inundifss  3408  relin1  4625  resss  4811  resmpt3  4836  cnvcnvss  4961  funin  5162  funimass2  5169  fnresin1  5205  fnres  5207  fresin  5269  ssimaex  5448  fneqeql2  5495  isoini2  5686  ofrfval  5956  ofvalg  5957  ofrval  5958  off  5960  ofres  5962  ofco  5966  smores  6155  smores2  6157  tfrlem5  6177  pmresg  6536  unfiin  6780  sbthlem7  6817  peano5nnnn  7664  peano5nni  8680  rexanuz  10700  fvsetsid  11888  tgvalex  12114  tgval2  12115  eltg3  12121  tgcl  12128  tgdom  12136  tgidm  12138  epttop  12154  ntropn  12181  ntrin  12188  cnptopresti  12302  cnptoprest  12303  txcnmpt  12337  xmetres  12446  metres  12447  blin2  12496  metrest  12570  tgioo  12610  limcresi  12678
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