Theorem snssgOLD 3803

Description: Obsolete version of snssgOLD 3803 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 2292	. 2 |- ( x = A -> ( x e. B <-> A e. B ) )
2		sneq 3677	. . 3 |- ( x = A -> { x } = { A } )
3	2	sseq1d 3253	. 2 |- ( x = A -> ( { x } C_ B <-> { A } C_ B ) )
4		vex 2802	. . 3 |- x e. _V
5	4	snss 3802	. 2 |- ( x e. B <-> { x } C_ B )
6	1, 3, 5	vtoclbg 2862	1 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   C_ wss 3197   {csn 3666

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-sn 3672

This theorem is referenced by: (None)