Theorem sseqin2 3444

Description: A relationship between subclass and intersection. Similar to Exercise 9 of [TakeutiZaring] p. 18. (Contributed by NM, 17-May-1994.)

Assertion

Ref	Expression
sseqin2	|- ( A C_ B <-> ( B i^i A ) = A )

Proof of Theorem sseqin2

Step	Hyp	Ref	Expression
1		dfss1 3429	1 |- ( A C_ B <-> ( B i^i A ) = A )

Syntax hints:   <-> wb 105   = wceq 1398   i^i cin 3213   C_ wss 3214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227

This theorem is referenced by:  dfss4st  3458  resabs1  5072  mptimass  5119  rescnvcnv  5230  fsuppeq  6460  fsuppeqg  6461  frecfnom  6645  fiintim  7204  nn0supp  9569  uzin  9905  iooval2  10267  fzval2  10364  suprzubdc  10620  bitsinv1  12673  dfphi2  12942  ballotfilemfmpn  13178  ressabsg  13373  resttopon  15162  restabs  15166  restopnb  15172  txcnmpt  15264