Theorem 0nep0 4297

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 4255	. . 3 |- (/) e. _V
2	1	snnz 3827	. 2 |- { (/) } =/= (/)
3	2	necomi 2505	1 |- (/) =/= { (/) }

Syntax hints:   =/= wne 2420   (/)c0 3520   {csn 3705

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4254

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-nul 3521  df-sn 3711

This theorem is referenced by:  0inp0  4298  opthprc  4821  2dom  7083  exmidpw  7205  exmidpw2en  7209  exmidaclem  7554  pw1dom2  7576