Theorem 0elpw 4254

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 3533	. 2 |- (/) C_ A
2		0ex 4216	. . 3 |- (/) e. _V
3	2	elpw 3658	. 2 |- ( (/) e. ~P A <-> (/) C_ A )
4	1, 3	mpbir 146	1 |- (/) e. ~P A

Syntax hints:   e. wcel 2202   C_ wss 3200   (/)c0 3494   ~Pcpw 3652

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654

This theorem is referenced by:  ordpwsucexmid  4668  pw1on  7443  pw1ne0  7445  pw1nct  16604