Theorem nosgnn0 27798

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 8471	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3013	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6446	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 3009	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8453	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3021	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 281	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 233	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2958	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 893	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5269	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4613	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 326	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 860   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∅c0 4285  {cpr 4590  suc csuc 6362  1_oc1o 8445  2_oc2o 8446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-v 3455  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4589  df-pr 4591  df-suc 6366  df-1o 8452  df-2o 8453

This theorem is referenced by:  nosgnn0i  27799  ltsres  27802  noseponlem  27804  ltsso  27816  nosepssdm  27826  nodenselem8  27831  nolt02olem  27834