Theorem nosgnn0i 33789

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 33788	. . 3 ⊢ ¬ ∅ ∈ {1o, 2o}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
3		eleq1 2826	. . . 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 2945	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∅c0 4253  {cpr 4560  1_oc1o 8260  2_oc2o 8261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-sn 4559  df-pr 4561  df-suc 6257  df-1o 8267  df-2o 8268

This theorem is referenced by:  sltres  33792  noextenddif  33798  nolesgn2ores  33802  nosepnelem  33809  nosepdmlem  33813  nolt02o  33825  nosupbnd1lem3  33840  nosupbnd1lem5  33842  nosupbnd2lem1  33845