Theorem snssg 3801

Description: The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)

Assertion

Ref	Expression
snssg	|- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Proof of Theorem snssg

Step	Hyp	Ref	Expression
1		snssb 3800	. . 3 |- ( { A } C_ B <-> ( A e. _V -> A e. B ) )
2	1	bicomi 132	. 2 |- ( ( A e. _V -> A e. B ) <-> { A } C_ B )
3		elex 2811	. 2 |- ( A e. V -> A e. _V )
4		imbibi 252	. 2 |- ( ( ( A e. _V -> A e. B ) <-> { A } C_ B ) -> ( A e. _V -> ( A e. B <-> { A } C_ B ) ) )
5	2, 3, 4	mpsyl 65	1 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   _Vcvv 2799   C_ wss 3197   {csn 3666

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-sn 3672

This theorem is referenced by:  snss  3802  snssi  3811  snssd  3812  prssg  3824  snelpwg  4295  ordtri2orexmid  4614  ordtri2or2exmid  4662  ontri2orexmidim  4663  relsng  4821  fvimacnvi  5748  fvimacnv  5749  tpfidceq  7088  strslfv  13072  strslfv3  13073  imasaddfnlemg  13342  imasaddvallemg  13343  lspsnid  14365  psrplusgg  14636  isneip  14814  elnei  14820  iscnp4  14886  cnpnei  14887  lpvtx  15873