Theorem snelpwg 4295

Description: A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) Put in closed form and avoid ax-nul 4209. (Revised by BJ, 17-Jan-2025.)

Assertion

Ref	Expression
snelpwg	|- ( A e. V -> ( A e. B <-> { A } e. ~P B ) )

Proof of Theorem snelpwg

Step	Hyp	Ref	Expression
1		snssg 3801	. 2 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )
2		snexg 4267	. . 3 |- ( A e. V -> { A } e. _V )
3		elpwg 3657	. . 3 |- ( { A } e. _V -> ( { A } e. ~P B <-> { A } C_ B ) )
4	2, 3	syl 14	. 2 |- ( A e. V -> ( { A } e. ~P B <-> { A } C_ B ) )
5	1, 4	bitr4d 191	1 |- ( A e. V -> ( A e. B <-> { A } e. ~P B ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2200   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   {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-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257

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-pw 3651  df-sn 3672

This theorem is referenced by: (None)