Theorem nosgnn0i 33599

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 33598	. . 3 ⊢ ¬ ∅ ∈ {1o, 2o}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
3		eleq1 2825	. . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
4	2, 3	mpbiri 261	. . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o})
5	1, 4	mto 200	. 2 ⊢ ¬ ∅ = 𝑋
6	5	neir 2943	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∅c0 4237  {cpr 4543  1_oc1o 8195  2_oc2o 8196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-nul 5199

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-sn 4542  df-pr 4544  df-suc 6219  df-1o 8202  df-2o 8203

This theorem is referenced by:  sltres  33602  noextenddif  33608  nolesgn2ores  33612  nosepnelem  33619  nosepdmlem  33623  nolt02o  33635  nosupbnd1lem3  33650  nosupbnd1lem5  33652  nosupbnd2lem1  33655