Theorem sseqin2 3440

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 3425	1 |- ( A C_ B <-> ( B i^i A ) = A )

Syntax hints:   <-> wb 105   = wceq 1398   i^i cin 3210   C_ wss 3211

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224

This theorem is referenced by:  dfss4st  3454  resabs1  5067  mptimass  5114  rescnvcnv  5225  fsuppeq  6447  fsuppeqg  6448  frecfnom  6632  fiintim  7191  nn0supp  9552  uzin  9887  iooval2  10248  fzval2  10345  suprzubdc  10596  bitsinv1  12648  dfphi2  12917  ressabsg  13289  resttopon  15036  restabs  15040  restopnb  15046  txcnmpt  15138