Theorem nosgnn0 28045

Description: ∅ is not a surreal sign. (Contributed by Scott Fenton, 16-Jun-2011.)

Assertion

Ref	Expression
nosgnn0	⊢ ¬ ∅ ∈ {1o, 2o}

Proof of Theorem nosgnn0

Step	Hyp	Ref	Expression
1		1n0 8523	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3016	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6466	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 3012	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8505	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3024	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 281	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 233	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2961	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 894	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5264	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4609	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 326	1 ⊢ ¬ ∅ ∈ {1o, 2o}

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281  {cpr 4590  suc csuc 6382  1_oc1o 8497  2_oc2o 8498

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5263

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-pr 4591  df-suc 6386  df-1o 8504  df-2o 8505

This theorem is used by:  nosgnn0i  28046  ltsres  28049  noseponlem  28051  ltsso  28063  nosepssdm  28073  nodenselem8  28078  nolt02olem  28081