Theorem snssg 3812

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 3811	. . 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 2815	. 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 2803   C_ wss 3201   {csn 3673

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-sn 3679

This theorem is referenced by:  snss  3813  snssi  3822  snssd  3823  prssg  3835  snelpwg  4308  ordtri2orexmid  4627  ordtri2or2exmid  4675  ontri2orexmidim  4676  relsng  4835  fvimacnvi  5770  fvimacnv  5771  tpfidceq  7165  strslfv  13190  strslfv3  13191  imasaddfnlemg  13460  imasaddvallemg  13461  lspsnid  14486  psrplusgg  14762  isneip  14940  elnei  14946  iscnp4  15012  cnpnei  15013  lpvtx  16003