Theorem snssgOLD 3835

Description: Obsolete version of snssgOLD 3835 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 2297	. 2 |- ( x = A -> ( x e. B <-> A e. B ) )
2		sneq 3705	. . 3 |- ( x = A -> { x } = { A } )
3	2	sseq1d 3271	. 2 |- ( x = A -> ( { x } C_ B <-> { A } C_ B ) )
4		vex 2818	. . 3 |- x e. _V
5	4	snss 3834	. 2 |- ( x e. B <-> { x } C_ B )
6	1, 3, 5	vtoclbg 2878	1 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2205   C_ wss 3214   {csn 3694

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227  df-sn 3700

This theorem is referenced by: (None)