Theorem snss 3813

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3812). 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 3812	. 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 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:  snssgOLD  3814  prss  3834  tpss  3846  snelpw  4310  sspwb  4314  mss  4324  exss  4325  reg2exmidlema  4638  elomssom  4709  relsn  4837  fnressn  5848  un0mulcl  9495  nn0ssz  9558  fimaxre2  11867  fsum2dlemstep  12075  fsumabs  12106  fsumiun  12118  fprod2dlemstep  12263  dvmptfsum  15536  elply2  15546  elplyd  15552  ply1term  15554  plymullem  15561  bdsnss  16589