Theorem 0inp0 2738

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 2737	. . 3 |- (/) =/= {(/)}
2		neeq1 1590	. . 3 |- (A = (/) -> (A =/= {(/)} <-> (/) =/= {(/)}))
3	1, 2	mpbiri 194	. 2 |- (A = (/) -> A =/= {(/)})
4		df-ne 1587	. 2 |- (A =/= {(/)} <-> -. A = {(/)})
5	3, 4	sylib 198	1 |- (A = (/) -> -. A = {(/)})

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   =/= wne 1585  (/)c0 2280  {csn 2409

This theorem is referenced by:  dtru 2772  zfpair 2777

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-nul 2710

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-nul 2281  df-sn 2412  df-pr 2413

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