Theorem nosgnn0i 14000

Description: If X is a surreal sign, then it is not null.

Hypothesis

Ref	Expression
nosgnn0i.1	|- X e. {1o, 2o}

Assertion

Ref	Expression
nosgnn0i	|- (/) =/= X

Proof of Theorem nosgnn0i

Step	Hyp	Ref	Expression
1		nosgnn0 13999	. . 3 |- -. (/) e. {1o, 2o}
2		nosgnn0i.1	. . . 4 |- X e. {1o, 2o}
3		eleq1 1957	. . . 4 |- ((/) = X -> ((/) e. {1o, 2o} <-> X e. {1o, 2o}))
4	2, 3	mpbiri 211	. . 3 |- ((/) = X -> (/) e. {1o, 2o})
5	1, 4	mto 121	. 2 |- -. (/) = X
6		df-ne 2019	. 2 |- ((/) =/= X <-> -. (/) = X)
7	5, 6	mpbir 207	1 |- (/) =/= X

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

This theorem is referenced by:  axfelem1 14031  axfelem3 14033  axfelem4 14034  axfelem5 14035  axfelem7 14037  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

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