Theorem snssgOLD 3768

Description: Obsolete version of snssgOLD 3768 as of 1-Jan-2025. (Contributed by NM, 22-Jul-2001.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem snssgOLD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2267	. 2 |- ( x = A -> ( x e. B <-> A e. B ) )
2		sneq 3643	. . 3 |- ( x = A -> { x } = { A } )
3	2	sseq1d 3221	. 2 |- ( x = A -> ( { x } C_ B <-> { A } C_ B ) )
4		vex 2774	. . 3 |- x e. _V
5	4	snss 3767	. 2 |- ( x e. B <-> { x } C_ B )
6	1, 3, 5	vtoclbg 2833	1 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   e. wcel 2175   C_ wss 3165   {csn 3632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-sn 3638

This theorem is referenced by: (None)