Theorem 0nep0 4225

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 4187	. . 3 |- (/) e. _V
2	1	snnz 3762	. 2 |- { (/) } =/= (/)
3	2	necomi 2463	1 |- (/) =/= { (/) }

Syntax hints:   =/= wne 2378   (/)c0 3468   {csn 3643

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-nul 4186

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-v 2778  df-dif 3176  df-nul 3469  df-sn 3649

This theorem is referenced by:  0inp0  4226  opthprc  4744  2dom  6921  exmidpw  7031  exmidpw2en  7035  exmidaclem  7351  pw1dom2  7373