Theorem snssg 3767

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 3766	. . 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 2783	. 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 2176   _Vcvv 2772   C_ wss 3166   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-sn 3639

This theorem is referenced by:  snss  3768  snssi  3777  snssd  3778  prssg  3790  ordtri2orexmid  4571  ordtri2or2exmid  4619  ontri2orexmidim  4620  relsng  4778  fvimacnvi  5694  fvimacnv  5695  tpfidceq  7027  strslfv  12877  strslfv3  12878  imasaddfnlemg  13146  imasaddvallemg  13147  lspsnid  14169  psrplusgg  14440  isneip  14618  elnei  14624  iscnp4  14690  cnpnei  14691