Theorem snssg 3500

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 2100	. 2 |- ( x = A -> ( x e. B <-> A e. B ) )
2		sneq 3386	. . 3 |- ( x = A -> { x } = { A } )
3	2	sseq1d 2972	. 2 |- ( x = A -> ( { x } C_ B <-> { A } C_ B ) )
4		vex 2560	. . 3 |- x e. _V
5	4	snss 3494	. 2 |- ( x e. B <-> { x } C_ B )
6	1, 3, 5	vtoclbg 2614	1 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   C_ wss 2917   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-sn 3381

This theorem is referenced by:  snssi  3508  snssd  3509  prssg  3521  ordtri2orexmid  4248  ordtri2or2exmid  4296  fvimacnvi  5281  fvimacnv  5282