Theorem snssb 3843

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 3235	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
2		velsn 3722	. . . 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 1523	. 2 |- ( A. x ( x e. { A } -> x e. B ) <-> A. x ( x = A -> x e. B ) )
5		eleq1 2301	. . . . 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 1523	. . 3 |- ( A. x ( x = A -> x e. B ) <-> A. x ( x = A -> A e. B ) )
8		19.23v 1936	. . 3 |- ( A. x ( x = A -> A e. B ) <-> ( E. x x = A -> A e. B ) )
9		isset 2828	. . . . 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 1400   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-sn 3711

This theorem is referenced by:  snssg  3844