Theorem nosgnn0i 27686

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 27685	. . 3 ⊢ ¬ ∅ ∈ {1o, 2o}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
3		eleq1 2814	. . . 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 2933	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∅c0 4322  {cpr 4625  1_oc1o 8481  2_oc2o 8482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5303

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-v 3464  df-dif 3949  df-un 3951  df-nul 4323  df-sn 4624  df-pr 4626  df-suc 6374  df-1o 8488  df-2o 8489

This theorem is referenced by:  sltres  27689  noextenddif  27695  nolesgn2ores  27699  nosepnelem  27706  nosepdmlem  27710  nolt02o  27722  nosupbnd1lem3  27737  nosupbnd1lem5  27739  nosupbnd2lem1  27742