Theorem snssg 3833

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 3832	. . 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 2827	. 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 2205   _Vcvv 2815   C_ wss 3214   {csn 3694

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227  df-sn 3700

This theorem is referenced by:  snss  3834  snssi  3843  snssd  3844  prssg  3856  snelpwg  4331  ordtri2orexmid  4650  ordtri2or2exmid  4698  ontri2orexmidim  4699  relsng  4858  fvimacnvi  5797  fvimacnv  5798  tpfidceq  7203  strslfv  13341  strslfv3  13342  imasaddfnlemg  13578  imasaddvallemg  13579  lspsnid  14681  psrplusgg  14959  isneip  15137  elnei  15143  iscnp4  15209  cnpnei  15210  lpvtx  16200