Theorem inss1 3218

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	⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴

Proof of Theorem inss1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3181	. . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
2	1	simplbi 268	. 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) → 𝑥 ∈ 𝐴)
3	2	ssriv 3027	1 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴

Syntax hints:   ∈ wcel 1438   ∩ cin 2996   ⊆ wss 2997

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010

This theorem is referenced by:  inss2  3219  ssinss1  3226  unabs  3228  inssddif  3238  inv1  3316  disjdif  3352  inundifss  3357  relin1  4543  resss  4724  resmpt3  4748  cnvcnvss  4872  funin  5071  funimass2  5078  fnresin1  5114  fnres  5116  fresin  5173  ssimaex  5349  fneqeql2  5392  isoini2  5580  ofrfval  5846  fnofval  5847  ofrval  5848  off  5850  ofres  5851  ofco  5855  smores  6039  smores2  6041  tfrlem5  6061  pmresg  6413  unfiin  6616  sbthlem7  6651  peano5nnnn  7406  peano5nni  8397  rexanuz  10385