Theorem snssg 3595

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 2157	. 2 |- ( x = A -> ( x e. B <-> A e. B ) )
2		sneq 3477	. . 3 |- ( x = A -> { x } = { A } )
3	2	sseq1d 3068	. 2 |- ( x = A -> ( { x } C_ B <-> { A } C_ B ) )
4		vex 2636	. . 3 |- x e. _V
5	4	snss 3588	. 2 |- ( x e. B <-> { x } C_ B )
6	1, 3, 5	vtoclbg 2694	1 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1296   e. wcel 1445   C_ wss 3013   {csn 3466

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-sn 3472

This theorem is referenced by:  snssi  3603  snssd  3604  prssg  3616  ordtri2orexmid  4367  ordtri2or2exmid  4415  relsng  4570  fvimacnvi  5452  fvimacnv  5453  strslfv  11703  isneip  12014  elnei  12020  iscnp4  12085  cnpnei  12086