Theorem 0nep0 4084

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 4050	. . 3 |- (/) e. _V
2	1	snnz 3637	. 2 |- { (/) } =/= (/)
3	2	necomi 2391	1 |- (/) =/= { (/) }

Syntax hints:   =/= wne 2306   (/)c0 3358   {csn 3522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-nul 4049

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-nul 3359  df-sn 3528

This theorem is referenced by:  0inp0  4085  opthprc  4585  2dom  6692  exmidpw  6795  exmidaclem  7057  pw1dom2  13179