Theorem snssgOLD 3775

Description: Obsolete version of snssgOLD 3775 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 2269	. 2 |- ( x = A -> ( x e. B <-> A e. B ) )
2		sneq 3649	. . 3 |- ( x = A -> { x } = { A } )
3	2	sseq1d 3226	. 2 |- ( x = A -> ( { x } C_ B <-> { A } C_ B ) )
4		vex 2776	. . 3 |- x e. _V
5	4	snss 3774	. 2 |- ( x e. B <-> { x } C_ B )
6	1, 3, 5	vtoclbg 2836	1 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2177   C_ wss 3170   {csn 3638

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183  df-sn 3644

This theorem is referenced by: (None)