Theorem snelpwi 4346

Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)

Assertion

Ref	Expression
snelpwi	|- ( A e. B -> { A } e. ~P B )

Proof of Theorem snelpwi

Step	Hyp	Ref	Expression
1		snssi 3854	. 2 |- ( A e. B -> { A } C_ B )
2		elex 2833	. . 3 |- ( A e. B -> A e. _V )
3		snexg 4316	. . 3 |- ( A e. _V -> { A } e. _V )
4		elpwg 3693	. . 3 |- ( { A } e. _V -> ( { A } e. ~P B <-> { A } C_ B ) )
5	2, 3, 4	3syl 17	. 2 |- ( A e. B -> ( { A } e. ~P B <-> { A } C_ B ) )
6	1, 5	mpbird 167	1 |- ( A e. B -> { A } e. ~P B )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685   {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711

This theorem is referenced by:  unipw  4352  infpwfidom  7540  txdis  15301  txdis1cn  15302