Theorem 0nep0 4209

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 4171	. . 3 |- (/) e. _V
2	1	snnz 3752	. 2 |- { (/) } =/= (/)
3	2	necomi 2461	1 |- (/) =/= { (/) }

Syntax hints:   =/= wne 2376   (/)c0 3460   {csn 3633

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-nul 4170

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-v 2774  df-dif 3168  df-nul 3461  df-sn 3639

This theorem is referenced by:  0inp0  4210  opthprc  4726  2dom  6897  exmidpw  7005  exmidpw2en  7009  exmidaclem  7320  pw1dom2  7339