Theorem nelsnOLD 4213

Description: Obsolete proof of nelsn 4212 as of 4-May-2021. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nelsnOLD	|- ( A =/= B -> -. A e. { B } )

Proof of Theorem nelsnOLD

Step	Hyp	Ref	Expression
1		neneq 2800	. . . 4 |- ( A =/= B -> -. A = B )
2	1	adantr 481	. . 3 |- ( ( A =/= B /\ A e. _V ) -> -. A = B )
3		elsng 4191	. . . 4 |- ( A e. _V -> ( A e. { B } <-> A = B ) )
4	3	adantl 482	. . 3 |- ( ( A =/= B /\ A e. _V ) -> ( A e. { B } <-> A = B ) )
5	2, 4	mtbird 315	. 2 |- ( ( A =/= B /\ A e. _V ) -> -. A e. { B } )
6		prcnel 3218	. . 3 |- ( -. A e. _V -> -. A e. { B } )
7	6	adantl 482	. 2 |- ( ( A =/= B /\ -. A e. _V ) -> -. A e. { B } )
8	5, 7	pm2.61dan 832	1 |- ( A =/= B -> -. A e. { B } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {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

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-sn 4178

This theorem is referenced by: (None)