Theorem 0nep0 2742

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 2716	. . 3 |- (/) e. V
2	1	snnz 2462	. 2 |- {(/)} =/= (/)
3		necom 1639	. 2 |- ({(/)} =/= (/) <-> (/) =/= {(/)})
4	2, 3	mpbi 189	1 |- (/) =/= {(/)}

Syntax hints:   =/= wne 1588  (/)c0 2283  {csn 2413

This theorem is referenced by:  0inp0 2743  opthprc 3227  2dom 4433

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-nul 2715

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-nul 2284  df-sn 2416  df-pr 2417

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