Theorem snss 3768

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3767). 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 3767	. 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 2176   _Vcvv 2772   C_ wss 3166   {csn 3633

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-sn 3639

This theorem is referenced by:  snssgOLD  3769  prss  3789  tpss  3799  snelpw  4258  sspwb  4261  mss  4271  exss  4272  reg2exmidlema  4583  elomssom  4654  relsn  4781  fnressn  5772  un0mulcl  9331  nn0ssz  9392  fimaxre2  11571  fsum2dlemstep  11778  fsumabs  11809  fsumiun  11821  fprod2dlemstep  11966  dvmptfsum  15230  elply2  15240  elplyd  15246  ply1term  15248  plymullem  15255  bdsnss  15846