Theorem 0inp0 4145

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 4144	. . 3 |- (/) =/= { (/) }
2		neeq1 2349	. . 3 |- ( A = (/) -> ( A =/= { (/) } <-> (/) =/= { (/) } ) )
3	1, 2	mpbiri 167	. 2 |- ( A = (/) -> A =/= { (/) } )
4	3	neneqd 2357	1 |- ( A = (/) -> -. A = { (/) } )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   =/= wne 2336   (/)c0 3409   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-nul 3410  df-sn 3582

This theorem is referenced by: (None)