Theorem inss1 3475

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 3219	. . 3 ⊢ (x ∈ (A ∩ B) ↔ (x ∈ A ∧ x ∈ B))
2	1	simplbi 446	. 2 ⊢ (x ∈ (A ∩ B) → x ∈ A)
3	2	ssriv 3277	1 ⊢ (A ∩ B) ⊆ A

Syntax hints:   ∈ wcel 1710   ∩ cin 3208   ⊆ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  inss2  3476  ssinss1  3483  unabs  3485  nssinpss  3487  dfin4  3495  inv1  3577  disjdif  3622  uniintsn  3963  pw1sspw  4171  inxpk  4277  cokrelk  4284  cnvkexg  4286  ssetkex  4294  sikexg  4296  ins2kexg  4305  ins3kexg  4306  dfidk2  4313  peano5  4409  phialllem2  4617  resss  4988  funin  5163  funimass2  5170  fnresin1  5197  fnres  5199  isoini2  5498  clos1induct  5880  erdisj  5972  sbthlem1  6203