Theorem snssg 3716

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 2233	. 2 |- ( x = A -> ( x e. B <-> A e. B ) )
2		sneq 3594	. . 3 |- ( x = A -> { x } = { A } )
3	2	sseq1d 3176	. 2 |- ( x = A -> ( { x } C_ B <-> { A } C_ B ) )
4		vex 2733	. . 3 |- x e. _V
5	4	snss 3709	. 2 |- ( x e. B <-> { x } C_ B )
6	1, 3, 5	vtoclbg 2791	1 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   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:  snssi  3724  snssd  3725  prssg  3737  ordtri2orexmid  4507  ordtri2or2exmid  4555  ontri2orexmidim  4556  relsng  4714  fvimacnvi  5610  fvimacnv  5611  strslfv  12460  isneip  12940  elnei  12946  iscnp4  13012  cnpnei  13013