Theorem snssg 3807

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 3806	. . 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 2814	. 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 2202   _Vcvv 2802   C_ wss 3200   {csn 3669

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-sn 3675

This theorem is referenced by:  snss  3808  snssi  3817  snssd  3818  prssg  3830  snelpwg  4302  ordtri2orexmid  4621  ordtri2or2exmid  4669  ontri2orexmidim  4670  relsng  4829  fvimacnvi  5761  fvimacnv  5762  tpfidceq  7121  strslfv  13126  strslfv3  13127  imasaddfnlemg  13396  imasaddvallemg  13397  lspsnid  14420  psrplusgg  14691  isneip  14869  elnei  14875  iscnp4  14941  cnpnei  14942  lpvtx  15929