Theorem snss 3838

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		elsn 3748	. . . 4 |- (x e. {A} <-> x = A)
2	1	imbi1i 315	. . 3 |- ((x e. {A} -> x e. B) <-> (x = A -> x e. B))
3	2	albii 1566	. 2 |- (A.x(x e. {A} -> x e. B) <-> A.x(x = A -> x e. B))
4		dfss2 3262	. 2 |- ({A} C_ B <-> A.x(x e. {A} -> x e. B))
5		snss.1	. . 3 |- A e. _V
6	5	clel2 2975	. 2 |- (A e. B <-> A.x(x = A -> x e. B))
7	3, 4, 6	3bitr4ri 269	1 |- (A e. B <-> {A} C_ B)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  _Vcvv 2859   C_ wss 3257  {csn 3737

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-sn 3741

This theorem is referenced by:  snssg  3844  prss  3861  tpss  3871  sspwb  4118  elpw1  4144  nnsucelrlem3  4426  nnsucelr  4428  tfinnn  4534  vfinspeqtncv  4553  brssetsn  4759  fvimacnvi  5402  fvimacnv  5403  fnressn  5438  dfnnc3  5885  mapsn  6026  nc0le1  6216  frecxp  6314