Theorem snssg 3651

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, 22-Jul-2001.)

Assertion

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

Proof of Theorem snssg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2200	. 2 |- ( x = A -> ( x e. B <-> A e. B ) )
2		sneq 3533	. . 3 |- ( x = A -> { x } = { A } )
3	2	sseq1d 3121	. 2 |- ( x = A -> ( { x } C_ B <-> { A } C_ B ) )
4		vex 2684	. . 3 |- x e. _V
5	4	snss 3644	. 2 |- ( x e. B <-> { x } C_ B )
6	1, 3, 5	vtoclbg 2742	1 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   C_ wss 3066   {csn 3522

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-sn 3528

This theorem is referenced by:  snssi  3659  snssd  3660  prssg  3672  ordtri2orexmid  4433  ordtri2or2exmid  4481  relsng  4637  fvimacnvi  5527  fvimacnv  5528  strslfv  11992  isneip  12304  elnei  12310  iscnp4  12376  cnpnei  12377