Theorem snelpw 3949

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. ~P B )

Proof of Theorem snelpw

Step	Hyp	Ref	Expression
1		snelpw.1	. . 3 |- A e. _V
2	1	snss 3494	. 2 |- ( A e. B <-> { A } C_ B )
3		snexgOLD 3935	. . . 4 |- ( A e. _V -> { A } e. _V )
4	1, 3	ax-mp 7	. . 3 |- { A } e. _V
5	4	elpw 3365	. 2 |- ( { A } e. ~P B <-> { A } C_ B )
6	2, 5	bitr4i 176	1 |- ( A e. B <-> { A } e. ~P B )

Syntax hints:   <-> wb 98   e. wcel 1393   _Vcvv 2557   C_ wss 2917   ~Pcpw 3359   {csn 3375

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381

This theorem is referenced by: (None)