Theorem 0nep0 2792

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 2766	. . 3 |- (/) e. V
2	1	snnz 2512	. 2 |- {(/)} =/= (/)
3		necom 1683	. 2 |- ({(/)} =/= (/) <-> (/) =/= {(/)})
4	2, 3	mpbi 196	1 |- (/) =/= {(/)}

Syntax hints:   =/= wne 1632  (/)c0 2331  {csn 2461

This theorem is referenced by:  0inp0 2793  opthprc 3278  2dom 4488

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-nul 2765

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-un 2101  df-nul 2332  df-sn 2464  df-pr 2465

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