Theorem 0inp0 2793

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 2792	. . 3 |- (/) =/= {(/)}
2		neeq1 1637	. . 3 |- (A = (/) -> (A =/= {(/)} <-> (/) =/= {(/)}))
3	1, 2	mpbiri 201	. 2 |- (A = (/) -> A =/= {(/)})
4		df-ne 1634	. 2 |- (A =/= {(/)} <-> -. A = {(/)})
5	3, 4	sylib 205	1 |- (A = (/) -> -. A = {(/)})

Syntax hints:  -. wn 2   -> wi 3   = wceq 997   =/= wne 1632  (/)c0 2331  {csn 2461

This theorem is referenced by:  dtru 2828  zfpair 2833

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-nul 2765

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-un 2101  df-nul 2332  df-sn 2464  df-pr 2465

Copyright terms: Public domain