Theorem 0inp0 4098

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 4097	. . 3 |- (/) =/= { (/) }
2		neeq1 2322	. . 3 |- ( A = (/) -> ( A =/= { (/) } <-> (/) =/= { (/) } ) )
3	1, 2	mpbiri 167	. 2 |- ( A = (/) -> A =/= { (/) } )
4	3	neneqd 2330	1 |- ( A = (/) -> -. A = { (/) } )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1332   =/= wne 2309   (/)c0 3368   {csn 3532

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3078  df-nul 3369  df-sn 3538

This theorem is referenced by: (None)