Theorem 0inp0 4837

Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 21-Jun-1993.)

Assertion

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

Proof of Theorem 0inp0

Step	Hyp	Ref	Expression
1		0nep0 4836	. . 3 |- (/) =/= { (/) }
2		neeq1 2856	. . 3 |- ( A = (/) -> ( A =/= { (/) } <-> (/) =/= { (/) } ) )
3	1, 2	mpbiri 248	. 2 |- ( A = (/) -> A =/= { (/) } )
4	3	neneqd 2799	1 |- ( A = (/) -> -. A = { (/) } )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   =/= wne 2794   (/)c0 3915   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916  df-sn 4178

This theorem is referenced by:  dtruALT  4899  zfpair  4904  dtruALT2  4911