Theorem nosgnn0i 33862

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 33861	. . 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 2946	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  {cpr 4563  1_oc1o 8290  2_oc2o 8291

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564  df-suc 6272  df-1o 8297  df-2o 8298

This theorem is referenced by:  sltres  33865  noextenddif  33871  nolesgn2ores  33875  nosepnelem  33882  nosepdmlem  33886  nolt02o  33898  nosupbnd1lem3  33913  nosupbnd1lem5  33915  nosupbnd2lem1  33918