Theorem 0elpw 4296

Description: Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)

Assertion

Ref	Expression
0elpw	|- (/) e. ~P A

Proof of Theorem 0elpw

Step	Hyp	Ref	Expression
1		0ss 3561	. 2 |- (/) C_ A
2		0ex 4255	. . 3 |- (/) e. _V
3	2	elpw 3691	. 2 |- ( (/) e. ~P A <-> (/) C_ A )
4	1, 3	mpbir 146	1 |- (/) e. ~P A

Syntax hints:   e. wcel 2209   C_ wss 3220   (/)c0 3520   ~Pcpw 3685

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-in1 623  ax-in2 624  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-ext 2220  ax-nul 4254

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-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687

This theorem is referenced by:  ordpwsucexmid  4712  pw1on  7575  pw1ne0  7577  ssenneg  11258  eupth2lemsfi  16633  pw1nct  16947  exmidpeirce  16951