Theorem nosgnn0i 27648

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 27647	. . 3 ⊢ ¬ ∅ ∈ {1o, 2o}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
3		eleq1 2828	. . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
4	2, 3	mpbiri 259	. . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o})
5	1, 4	mto 198	. 2 ⊢ ¬ ∅ = 𝑋
6	5	neir 2938	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∅c0 4268  {cpr 4564  1_oc1o 8395  2_oc2o 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-v 3434  df-dif 3893  df-un 3895  df-nul 4269  df-sn 4563  df-pr 4565  df-suc 6323  df-1o 8402  df-2o 8403

This theorem is referenced by:  ltsres  27651  noextenddif  27657  nolesgn2ores  27661  nosepnelem  27668  nosepdmlem  27672  nolt02o  27684  nosupbnd1lem3  27699  nosupbnd1lem5  27701  nosupbnd2lem1  27704