Theorem 0inp0 3999

Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)

Assertion

Ref	Expression
0inp0	|- ( A = (/) -> -. A = { (/) } )

Proof of Theorem 0inp0

Step	Hyp	Ref	Expression
1		0nep0 3998	. . 3 |- (/) =/= { (/) }
2		neeq1 2268	. . 3 |- ( A = (/) -> ( A =/= { (/) } <-> (/) =/= { (/) } ) )
3	1, 2	mpbiri 166	. 2 |- ( A = (/) -> A =/= { (/) } )
4	3	neneqd 2276	1 |- ( A = (/) -> -. A = { (/) } )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1289   =/= wne 2255   (/)c0 3286   {csn 3444

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3963

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 3001  df-nul 3287  df-sn 3450

This theorem is referenced by: (None)