Theorem snss 3829

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3828). Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 21-Jun-1993.) (Proof shortened by BJ, 1-Jan-2025.)

Hypothesis

Ref	Expression
snss.1	|- A e. _V

Assertion

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

Proof of Theorem snss

Step	Hyp	Ref	Expression
1		snss.1	. 2 |- A e. _V
2		snssg 3828	. 2 |- ( A e. _V -> ( A e. B <-> { A } C_ B ) )
3	1, 2	ax-mp 5	1 |- ( A e. B <-> { A } C_ B )

Syntax hints:   <-> wb 105   e. wcel 2203   _Vcvv 2813   C_ wss 3211   {csn 3689

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-sn 3695

This theorem is referenced by:  snssgOLD  3830  prss  3850  tpss  3862  snelpw  4328  sspwb  4332  mss  4342  exss  4343  reg2exmidlema  4656  elomssom  4727  relsn  4855  fnressn  5870  un0mulcl  9530  nn0ssz  9595  hashfibclem  11206  fimaxre2  11912  fsum2dlemstep  12120  fsumabs  12151  fsumiun  12163  fprod2dlemstep  12308  dvmptfsum  15590  elply2  15600  elplyd  15606  ply1term  15608  plymullem  15615  bdsnss  16643