Theorem snss 3845

Description: The singleton of an element of a class is a subset of the class (inference form of snssg 3844). 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 3844	. 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 2209   _Vcvv 2821   C_ wss 3220   {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-sn 3711

This theorem is referenced by:  snssgOLD  3846  prss  3866  tpss  3878  snelpw  4347  sspwb  4351  mss  4361  exss  4362  reg2exmidlema  4676  elomssom  4747  relsn  4875  fnressn  5892  un0mulcl  9576  nn0ssz  9641  hashfibclem  11260  hashf1lem1  11263  hashf1lem2  11264  fimaxre2  11971  fsum2dlemstep  12179  fsumabs  12210  fsumiun  12222  fprod2dlemstep  12367  dvmptfsum  15749  elply2  15759  elplyd  15765  ply1term  15767  plymullem  15774  bdsnss  16813