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Theorem inss1 3151
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 3120 . . 3 (x (AB) ↔ (x A x B))
21simplbi 259 . 2 (x (AB) → x A)
32ssriv 2943 1 (AB) ⊆ A
Colors of variables: wff set class
Syntax hints:   wcel 1390  cin 2910  wss 2911
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925
This theorem is referenced by:  inss2  3152  ssinss1  3159  unabs  3161  nssinpss  3163  inssddif  3172  inv1  3247  disjdif  3290  inundifss  3295  relin1  4398  resss  4578  resmpt3  4600  cnvcnvss  4718  funin  4913  funimass2  4920  fnresin1  4956  fnres  4958  fresin  5011  ssimaex  5177  fneqeql2  5219  isoini2  5401  ofrfval  5662  fnofval  5663  ofrval  5664  off  5666  ofres  5667  ofco  5671  smores  5848  smores2  5850  tfrlem5  5871  peano5nni  7678
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