Theorem snss 3657

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 3549	. . . 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 1447	. 2 |- ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
4		dfss2 3091	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
5		snss.1	. . 3 |- A e. _V
6	5	clel2 2822	. 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 1330   = wceq 1332   e. wcel 1481   _Vcvv 2689   C_ wss 3076   {csn 3532

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-sn 3538

This theorem is referenced by:  snssg  3664  prss  3684  tpss  3693  snelpw  4143  sspwb  4146  mss  4156  exss  4157  reg2exmidlema  4457  elnn  4527  relsn  4652  fnressn  5614  un0mulcl  9035  nn0ssz  9096  fimaxre2  11030  fsum2dlemstep  11235  fsumabs  11266  fsumiun  11278  bdsnss  13242