Theorem sseqin2 3450

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 3435	1 |- ( A C_ B <-> ( B i^i A ) = A )

Syntax hints:   <-> wb 105   = wceq 1402   i^i cin 3219   C_ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233

This theorem is referenced by:  dfss4st  3464  resabs1  5087  mptimass  5134  rescnvcnv  5245  fsuppeq  6477  fsuppeqg  6478  frecfnom  6662  fiintim  7228  nn0supp  9598  uzin  9934  iooval2  10296  fzval2  10393  suprzubdc  10649  bitsinv1  12707  dfphi2  12976  ballotfilemfmpn  13212  ressabsg  13407  resttopon  15195  restabs  15199  restopnb  15205  txcnmpt  15297