Theorem snelpwg 4308

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 4220. (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 3812	. 2 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )
2		snexg 4280	. . 3 |- ( A e. V -> { A } e. _V )
3		elpwg 3664	. . 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 2202   _Vcvv 2803   C_ wss 3201   ~Pcpw 3656   {csn 3673

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679

This theorem is referenced by: (None)