Theorem snelpwg 4267

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 4181. (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 3773	. 2 |- ( A e. V -> ( A e. B <-> { A } C_ B ) )
2		snexg 4239	. . 3 |- ( A e. V -> { A } e. _V )
3		elpwg 3629	. . 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 2177   _Vcvv 2773   C_ wss 3170   ~Pcpw 3621   {csn 3638

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644

This theorem is referenced by: (None)