Theorem sseqin2 3290

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 3275	1 |- ( A C_ B <-> ( B i^i A ) = A )

Syntax hints:   <-> wb 104   = wceq 1331   i^i cin 3065   C_ wss 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079

This theorem is referenced by:  dfss4st  3304  resabs1  4843  rescnvcnv  4996  frecfnom  6291  fiintim  6810  nn0supp  9022  uzin  9351  iooval2  9691  fzval2  9786  dfphi2  11885  resttopon  12329  restabs  12333  restopnb  12339  txcnmpt  12431