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  4257  sspwb  4260  mss  4270  exss  4271  reg2exmidlema  4582  elomssom  4653  relsn  4780  fnressn  5770  un0mulcl  9329  nn0ssz  9390  fimaxre2  11538  fsum2dlemstep  11745  fsumabs  11776  fsumiun  11788  fprod2dlemstep  11933  dvmptfsum  15197  elply2  15207  elplyd  15213  ply1term  15215  plymullem  15222  bdsnss  15809