Theorem snss 3709

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 3600	. . . 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 1463	. 2 |- ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
4		dfss2 3136	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
5		snss.1	. . 3 |- A e. _V
6	5	clel2 2863	. 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 1346   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   {csn 3583

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589

This theorem is referenced by:  snssg  3716  prss  3736  tpss  3745  snelpw  4198  sspwb  4201  mss  4211  exss  4212  reg2exmidlema  4518  elomssom  4589  relsn  4716  fnressn  5682  un0mulcl  9169  nn0ssz  9230  fimaxre2  11190  fsum2dlemstep  11397  fsumabs  11428  fsumiun  11440  fprod2dlemstep  11585  bdsnss  13908