Theorem snss 3803

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3802). 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 3802	. 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 2200   _Vcvv 2799   C_ wss 3197   {csn 3666

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-sn 3672

This theorem is referenced by:  snssgOLD  3804  prss  3824  tpss  3836  snelpw  4298  sspwb  4302  mss  4312  exss  4313  reg2exmidlema  4626  elomssom  4697  relsn  4824  fnressn  5825  un0mulcl  9403  nn0ssz  9464  fimaxre2  11738  fsum2dlemstep  11945  fsumabs  11976  fsumiun  11988  fprod2dlemstep  12133  dvmptfsum  15399  elply2  15409  elplyd  15415  ply1term  15417  plymullem  15424  bdsnss  16236