Theorem snssOLD 3770

Description: Obsolete version of snss 3779 as of 1-Jan-2025. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
snssOLD.1	|- A e. _V

Assertion

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

Proof of Theorem snssOLD

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 3660	. . . 4 |- ( x e. { A } <-> x = A )
2	1	imbi1i 238	. . 3 |- ( ( x e. { A } -> x e. B ) <-> ( x = A -> x e. B ) )
3	2	albii 1494	. 2 |- ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
4		ssalel 3189	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
5		snssOLD.1	. . 3 |- A e. _V
6	5	clel2 2913	. 2 |- ( A e. B <-> A. x ( x = A -> x e. B ) )
7	3, 4, 6	3bitr4ri 213	1 |- ( A e. B <-> { A } C_ B )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   {csn 3643

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 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-sn 3649

This theorem is referenced by: (None)