Theorem nosgnn0 13999

Description: (/) is not a surreal sign.

Assertion

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

Proof of Theorem nosgnn0

Step	Hyp	Ref	Expression
1		1n0 5187	. . . 4 |- 1o =/= (/)
2		necom 2094	. . . . 5 |- (1o =/= (/) <-> (/) =/= 1o)
3		df-ne 2019	. . . . 5 |- ((/) =/= 1o <-> -. (/) = 1o)
4	2, 3	bitri 190	. . . 4 |- (1o =/= (/) <-> -. (/) = 1o)
5	1, 4	mpbi 206	. . 3 |- -. (/) = 1o
6		nsuceq0 3749	. . . . 5 |- suc 1o =/= (/)
7		necom 2094	. . . . . 6 |- (suc 1o =/= (/) <-> (/) =/= suc 1o)
8		df-2o 5178	. . . . . . 7 |- 2o = suc 1o
9	8	neeq2i 2027	. . . . . 6 |- ((/) =/= 2o <-> (/) =/= suc 1o)
10	7, 9	bitr4i 193	. . . . 5 |- (suc 1o =/= (/) <-> (/) =/= 2o)
11	6, 10	mpbi 206	. . . 4 |- (/) =/= 2o
12		df-ne 2019	. . . 4 |- ((/) =/= 2o <-> -. (/) = 2o)
13	11, 12	mpbi 206	. . 3 |- -. (/) = 2o
14	5, 13	pm3.2ni 640	. 2 |- -. ((/) = 1o \/ (/) = 2o)
15		0ex 3446	. . 3 |- (/) e. _V
16	15	elpr 3061	. 2 |- ((/) e. {1o, 2o} <-> ((/) = 1o \/ (/) = 2o))
17	14, 16	mtbir 209	1 |- -. (/) e. {1o, 2o}

Syntax hints:  -. wn 2   \/ wo 239   = wceq 1298   e. wcel 1300   =/= wne 2017  (/)c0 2875  {cpr 3045  suc csuc 3659  1oc1o 5172  2oc2o 5173

This theorem is referenced by:  nosgnn0i 14000  axsltso 14007  axdenselem3 14021  axdenselem8 14026  axfelem12 14042

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-nul 3445

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-nul 2876  df-sn 3049  df-pr 3050  df-suc 3663  df-1o 5177  df-2o 5178

Copyright terms: Public domain