Theorem nosgnn0 31811

Description: (/) is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)

Assertion

Ref	Expression
nosgnn0	|- -. (/) e. { 1o , 2o }

Proof of Theorem nosgnn0

Step	Hyp	Ref	Expression
1		1n0 7575	. . . 4 |- 1o =/= (/)
2	1	nesymi 2851	. . 3 |- -. (/) = 1o
3		nsuceq0 5805	. . . . 5 |- suc 1o =/= (/)
4		necom 2847	. . . . . 6 |- ( suc 1o =/= (/) <-> (/) =/= suc 1o )
5		df-2o 7561	. . . . . . 7 |- 2o = suc 1o
6	5	neeq2i 2859	. . . . . 6 |- ( (/) =/= 2o <-> (/) =/= suc 1o )
7	4, 6	bitr4i 267	. . . . 5 |- ( suc 1o =/= (/) <-> (/) =/= 2o )
8	3, 7	mpbi 220	. . . 4 |- (/) =/= 2o
9	8	neii 2796	. . 3 |- -. (/) = 2o
10	2, 9	pm3.2ni 899	. 2 |- -. ( (/) = 1o \/ (/) = 2o )
11		0ex 4790	. . 3 |- (/) e. _V
12	11	elpr 4198	. 2 |- ( (/) e. { 1o , 2o } <-> ( (/) = 1o \/ (/) = 2o ) )
13	10, 12	mtbir 313	1 |- -. (/) e. { 1o , 2o }

Syntax hints:   -. wn 3   \/ wo 383   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   {cpr 4179   suc csuc 5725   1oc1o 7553   2oc2o 7554

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-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-suc 5729  df-1o 7560  df-2o 7561

This theorem is referenced by:  nosgnn0i  31812  sltres  31815  noseponlem  31817  sltso  31827  nosepssdm  31836  nodenselem8  31841  nolt02olem  31844