Theorem 0elpw 2731

Description: Every power class contains the empty set.

Assertion

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

Proof of Theorem 0elpw

Step	Hyp	Ref	Expression
1		0ss 2297	. 2 |- (/) (_ A
2		0ex 2706	. . 3 |- (/) e. V
3	2	elpw 2400	. 2 |- ((/) e. P~A <-> (/) (_ A)
4	1, 3	mpbir 190	1 |- (/) e. P~A

Syntax hints:   e. wcel 956   (_ wss 2043  (/)c0 2276  P~cpw 2397

This theorem is referenced by:  bcth 7982

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-nul 2705

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398

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