Theorem 0nep0 2727

Description: The empty set and its power set are not equal.

Assertion

Ref	Expression
0nep0	|- (/) =/= {(/)}

Proof of Theorem 0nep0

Step	Hyp	Ref	Expression
1		0ex 2701	. . 3 |- (/) e. V
2	1	snnz 2449	. 2 |- {(/)} =/= (/)
3		necom 1628	. 2 |- ({(/)} =/= (/) <-> (/) =/= {(/)})
4	2, 3	mpbi 189	1 |- (/) =/= {(/)}

Syntax hints:   =/= wne 1577  (/)c0 2270  {csn 2399

This theorem is referenced by:  0inp0 2728  opthprc 3211  2dom 4408

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-nul 2700

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-nul 2271  df-sn 2402  df-pr 2403

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