Theorem snelpwg 4302

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 4215. (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 3807	. 2 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )
2		snexg 4274	. . 3 |- ( A e. V -> { A } e. _V )
3		elpwg 3660	. . 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 2802   C_ wss 3200   ~Pcpw 3652   {csn 3669

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675

This theorem is referenced by: (None)