Theorem snelpw 3940

Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)

Hypothesis

Ref	Expression
snelpw.1	|- A e. _V

Assertion

Ref	Expression
snelpw	|- (A e. B <-> {A } e. ~PB)

Proof of Theorem snelpw

Step	Hyp	Ref	Expression
1		snelpw.1	. . 3 |- A e. _V
2	1	snss 3485	. 2 |- (A e. B <-> {A } C_ B)
3		snexgOLD 3926	. . . 4 |- (A e. _V -> {A } e. _V)
4	1, 3	ax-mp 7	. . 3 |- {A } e. _V
5	4	elpw 3357	. 2 |- ( {A } e. ~PB <-> {A } C_ B)
6	2, 5	bitr4i 176	1 |- (A e. B <-> {A } e. ~PB)

Syntax hints:   <-> wb 98   e. wcel 1390   _Vcvv 2551   C_ wss 2911   ~Pcpw 3351   {csn 3367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373

This theorem is referenced by: (None)