Theorem nosgnn0i 27642

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 27641	. . 3 ⊢ ¬ ∅ ∈ {1o, 2o}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
3		eleq1 2825	. . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
4	2, 3	mpbiri 258	. . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o})
5	1, 4	mto 197	. 2 ⊢ ¬ ∅ = 𝑋
6	5	neir 2936	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∅c0 4274  {cpr 4570  1_oc1o 8389  2_oc2o 8390

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3432  df-dif 3893  df-un 3895  df-nul 4275  df-sn 4569  df-pr 4571  df-suc 6321  df-1o 8396  df-2o 8397

This theorem is referenced by:  ltsres  27645  noextenddif  27651  nolesgn2ores  27655  nosepnelem  27662  nosepdmlem  27666  nolt02o  27678  nosupbnd1lem3  27693  nosupbnd1lem5  27695  nosupbnd2lem1  27698