Theorem sseqin2 3428

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 3413	1 |- ( A C_ B <-> ( B i^i A ) = A )

Syntax hints:   <-> wb 105   = wceq 1398   i^i cin 3200   C_ wss 3201

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214

This theorem is referenced by:  dfss4st  3442  resabs1  5048  mptimass  5095  rescnvcnv  5206  fsuppeq  6425  fsuppeqg  6426  frecfnom  6610  fiintim  7166  nn0supp  9498  uzin  9833  iooval2  10194  fzval2  10291  suprzubdc  10542  bitsinv1  12586  dfphi2  12855  ressabsg  13222  resttopon  14965  restabs  14969  restopnb  14975  txcnmpt  15067