Theorem nosgnn0i 28017

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 28016	. . 3 ⊢ ¬ ∅ ∈ {1o, 2o}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
3		eleq1 2848	. . . 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 2958	1 ⊢ ∅ ≠ 𝑋

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∅c0 4278  {cpr 4585  1_oc1o 8470  2_oc2o 8471

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4584  df-pr 4586  df-suc 6368  df-1o 8477  df-2o 8478

This theorem is used by:  ltsres  28020  noextenddif  28026  nolesgn2ores  28030  nosepnelem  28037  nosepdmlem  28041  nolt02o  28053  nosupbnd1lem3  28068  nosupbnd1lem5  28070  nosupbnd2lem1  28073