Theorem snss 3685

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 3577	. . . 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 1450	. 2 |- ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
4		dfss2 3117	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
5		snss.1	. . 3 |- A e. _V
6	5	clel2 2845	. 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 1333   = wceq 1335   e. wcel 2128   _Vcvv 2712   C_ wss 3102   {csn 3560

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-sn 3566

This theorem is referenced by:  snssg  3692  prss  3712  tpss  3721  snelpw  4173  sspwb  4176  mss  4186  exss  4187  reg2exmidlema  4492  elomssom  4563  relsn  4690  fnressn  5652  un0mulcl  9119  nn0ssz  9180  fimaxre2  11121  fsum2dlemstep  11326  fsumabs  11357  fsumiun  11369  fprod2dlemstep  11514  bdsnss  13435