Theorem bj-0nel1 32940

Description: The empty set does not belong to { 1o }. (Contributed by BJ, 6-Apr-2019.)

Assertion

Ref	Expression
bj-0nel1	|- (/) e/ { 1o }

Proof of Theorem bj-0nel1

Step	Hyp	Ref	Expression
1		1n0 7575	. . . 4 |- 1o =/= (/)
2	1	nesymi 2851	. . 3 |- -. (/) = 1o
3		0ex 4790	. . . 4 |- (/) e. _V
4	3	elsn 4192	. . 3 |- ( (/) e. { 1o } <-> (/) = 1o )
5	2, 4	mtbir 313	. 2 |- -. (/) e. { 1o }
6	5	nelir 2900	1 |- (/) e/ { 1o }

Syntax hints:   = wceq 1483   e. wcel 1990   e/ wnel 2897   (/)c0 3915   {csn 4177   1oc1o 7553

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-nel 2898  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-suc 5729  df-1o 7560

This theorem is referenced by: (None)