Theorem 0inp0 2728

Description: Something cannot be equal to both the null set and the power set of the null set.

Assertion

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

Proof of Theorem 0inp0

Step	Hyp	Ref	Expression
1		0nep0 2727	. . 3 |- (/) =/= {(/)}
2		neeq1 1582	. . 3 |- (A = (/) -> (A =/= {(/)} <-> (/) =/= {(/)}))
3	1, 2	mpbiri 194	. 2 |- (A = (/) -> A =/= {(/)})
4		df-ne 1579	. 2 |- (A =/= {(/)} <-> -. A = {(/)})
5	3, 4	sylib 198	1 |- (A = (/) -> -. A = {(/)})

Syntax hints:  -. wn 2   -> wi 3   = wceq 953   =/= wne 1577  (/)c0 2270  {csn 2399

This theorem is referenced by:  dtru 2762  zfpair 2767

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-nul 2700

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-nul 2271  df-sn 2402  df-pr 2403

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