Theorem snssOLD 3718

Description: Obsolete version of snss 3727 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 3609	. . . 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 1470	. 2 |- ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
4		dfss2 3144	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
5		snssOLD.1	. . 3 |- A e. _V
6	5	clel2 2870	. 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 1351   = wceq 1353   e. wcel 2148   _Vcvv 2737   C_ wss 3129   {csn 3592

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-sn 3598

This theorem is referenced by: (None)