Theorem nosgnn0i 27169

Description: If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)

Hypothesis

Ref	Expression
nosgnn0i.1	⊢ 𝑋 ∈ {1o, 2o}

Assertion

Ref	Expression
nosgnn0i	⊢ ∅ ≠ 𝑋

Proof of Theorem nosgnn0i

Step	Hyp	Ref	Expression
1		nosgnn0 27168	. . 3 ⊢ ¬ ∅ ∈ {1o, 2o}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
3		eleq1 2821	. . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
4	2, 3	mpbiri 257	. . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o})
5	1, 4	mto 196	. 2 ⊢ ¬ ∅ = 𝑋
6	5	neir 2943	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∅c0 4322  {cpr 4630  1_oc1o 8461  2_oc2o 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631  df-suc 6370  df-1o 8468  df-2o 8469

This theorem is referenced by:  sltres  27172  noextenddif  27178  nolesgn2ores  27182  nosepnelem  27189  nosepdmlem  27193  nolt02o  27205  nosupbnd1lem3  27220  nosupbnd1lem5  27222  nosupbnd2lem1  27225