Theorem sseqin2 3382

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 3367	1 |- ( A C_ B <-> ( B i^i A ) = A )

Syntax hints:   <-> wb 105   = wceq 1364   i^i cin 3156   C_ wss 3157

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170

This theorem is referenced by:  dfss4st  3396  resabs1  4975  mptimass  5022  rescnvcnv  5132  frecfnom  6459  fiintim  6992  nn0supp  9301  uzin  9634  iooval2  9990  fzval2  10086  suprzubdc  10326  dfphi2  12388  ressabsg  12754  resttopon  14407  restabs  14411  restopnb  14417  txcnmpt  14509