Theorem 0nep0 4836

Description: The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)

Assertion

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

Proof of Theorem 0nep0

Step	Hyp	Ref	Expression
1		0ex 4790	. . 3 |- (/) e. _V
2	1	snnz 4309	. 2 |- { (/) } =/= (/)
3	2	necomi 2848	1 |- (/) =/= { (/) }

Syntax hints:   =/= wne 2794   (/)c0 3915   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916  df-sn 4178

This theorem is referenced by:  0inp0  4837  opthprc  5167  2dom  8029  pw2eng  8066  hashge3el3dif  13268  isusp  22065  bj-1upln0  32997  clsk1indlem0  38339