Theorem 0inp0 4199

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 4198	. . 3 |- (/) =/= { (/) }
2		neeq1 2380	. . 3 |- ( A = (/) -> ( A =/= { (/) } <-> (/) =/= { (/) } ) )
3	1, 2	mpbiri 168	. 2 |- ( A = (/) -> A =/= { (/) } )
4	3	neneqd 2388	1 |- ( A = (/) -> -. A = { (/) } )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   =/= wne 2367   (/)c0 3450   {csn 3622

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4159

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-nul 3451  df-sn 3628

This theorem is referenced by: (None)