Theorem snss 3702

Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
snss.1	|- A e. _V

Assertion

Ref	Expression
snss	|- ( A e. B <-> { A } C_ B )

Proof of Theorem snss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 3593	. . . 4 |- ( x e. { A } <-> x = A )
2	1	imbi1i 237	. . 3 |- ( ( x e. { A } -> x e. B ) <-> ( x = A -> x e. B ) )
3	2	albii 1458	. 2 |- ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
4		dfss2 3131	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
5		snss.1	. . 3 |- A e. _V
6	5	clel2 2859	. 2 |- ( A e. B <-> A. x ( x = A -> x e. B ) )
7	3, 4, 6	3bitr4ri 212	1 |- ( A e. B <-> { A } C_ B )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-sn 3582

This theorem is referenced by:  snssg  3709  prss  3729  tpss  3738  snelpw  4191  sspwb  4194  mss  4204  exss  4205  reg2exmidlema  4511  elomssom  4582  relsn  4709  fnressn  5671  un0mulcl  9148  nn0ssz  9209  fimaxre2  11168  fsum2dlemstep  11375  fsumabs  11406  fsumiun  11418  fprod2dlemstep  11563  bdsnss  13765