Theorem 0nep0 4249

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 4211	. . 3 |- (/) e. _V
2	1	snnz 3786	. 2 |- { (/) } =/= (/)
3	2	necomi 2485	1 |- (/) =/= { (/) }

Syntax hints:   =/= wne 2400   (/)c0 3491   {csn 3666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4210

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-nul 3492  df-sn 3672

This theorem is referenced by:  0inp0  4250  opthprc  4770  2dom  6958  exmidpw  7070  exmidpw2en  7074  exmidaclem  7390  pw1dom2  7412