Theorem nosgnn0i 27162

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 27161	. . 3 ⊢ ¬ ∅ ∈ {1o, 2o}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
3		eleq1 2822	. . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
4	2, 3	mpbiri 258	. . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o})
5	1, 4	mto 196	. 2 ⊢ ¬ ∅ = 𝑋
6	5	neir 2944	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∅c0 4323  {cpr 4631  1_oc1o 8459  2_oc2o 8460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-v 3477  df-dif 3952  df-un 3954  df-nul 4324  df-sn 4630  df-pr 4632  df-suc 6371  df-1o 8466  df-2o 8467

This theorem is referenced by:  sltres  27165  noextenddif  27171  nolesgn2ores  27175  nosepnelem  27182  nosepdmlem  27186  nolt02o  27198  nosupbnd1lem3  27213  nosupbnd1lem5  27215  nosupbnd2lem1  27218