Theorem nosgnn0i 27950

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 27949	. . 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 1570   ∈ wcel 2145   ≠ wne 2955  ∅c0 4278  {cpr 4585  1_oc1o 8447  2_oc2o 8448

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452  df-dif 3901  df-un 3903  df-nul 4279  df-sn 4584  df-pr 4586  df-suc 6357  df-1o 8454  df-2o 8455

This theorem is used by:  ltsres  27953  noextenddif  27959  nolesgn2ores  27963  nosepnelem  27970  nosepdmlem  27974  nolt02o  27986  nosupbnd1lem3  28001  nosupbnd1lem5  28003  nosupbnd2lem1  28006