Theorem snss 3779

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3778). 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 3778	. 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 2178   _Vcvv 2776   C_ wss 3174   {csn 3643

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-sn 3649

This theorem is referenced by:  snssgOLD  3780  prss  3800  tpss  3812  snelpw  4274  sspwb  4278  mss  4288  exss  4289  reg2exmidlema  4600  elomssom  4671  relsn  4798  fnressn  5793  un0mulcl  9364  nn0ssz  9425  fimaxre2  11653  fsum2dlemstep  11860  fsumabs  11891  fsumiun  11903  fprod2dlemstep  12048  dvmptfsum  15312  elply2  15322  elplyd  15328  ply1term  15330  plymullem  15337  bdsnss  16008