Theorem 0inp0 3475

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 3474	. . 3 |- (/) =/= {(/)}
2		neeq1 2024	. . 3 |- (A = (/) -> (A =/= {(/)} <-> (/) =/= {(/)}))
3	1, 2	mpbiri 211	. 2 |- (A = (/) -> A =/= {(/)})
4		df-ne 2019	. 2 |- (A =/= {(/)} <-> -. A = {(/)})
5	3, 4	sylib 215	1 |- (A = (/) -> -. A = {(/)})

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   =/= wne 2017  (/)c0 2875  {csn 3044

This theorem is referenced by:  dtruALT 3517  zfpair 3522

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-nul 3445

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-nul 2876  df-sn 3049

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