Theorem snssb 3801

Description: Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)

Assertion

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

Proof of Theorem snssb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssalel 3212	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
2		velsn 3683	. . . 4 |- ( x e. { A } <-> x = A )
3	2	imbi1i 238	. . 3 |- ( ( x e. { A } -> x e. B ) <-> ( x = A -> x e. B ) )
4	3	albii 1516	. 2 |- ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
5		eleq1 2292	. . . . 5 |- ( x = A -> ( x e. B <-> A e. B ) )
6	5	pm5.74i 180	. . . 4 |- ( ( x = A -> x e. B ) <-> ( x = A -> A e. B ) )
7	6	albii 1516	. . 3 |- ( A. x ( x = A -> x e. B ) <-> A. x ( x = A -> A e. B ) )
8		19.23v 1929	. . 3 |- ( A. x ( x = A -> A e. B ) <-> ( E. x x = A -> A e. B ) )
9		isset 2806	. . . . 5 |- ( A e. _V <-> E. x x = A )
10	9	bicomi 132	. . . 4 |- ( E. x x = A <-> A e. _V )
11	10	imbi1i 238	. . 3 |- ( ( E. x x = A -> A e. B ) <-> ( A e. _V -> A e. B ) )
12	7, 8, 11	3bitri 206	. 2 |- ( A. x ( x = A -> x e. B ) <-> ( A e. _V -> A e. B ) )
13	1, 4, 12	3bitri 206	1 |- ( { A } C_ B <-> ( A e. _V -> A e. B ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   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:  snssg  3802