Theorem nosgnn0 33788

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 8286	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3000	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6331	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 2996	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8268	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3008	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 277	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 229	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2944	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 877	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5226	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4581	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 322	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∅c0 4253  {cpr 4560  suc csuc 6253  1_oc1o 8260  2_oc2o 8261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-sn 4559  df-pr 4561  df-suc 6257  df-1o 8267  df-2o 8268

This theorem is referenced by:  nosgnn0i  33789  sltres  33792  noseponlem  33794  sltso  33806  nosepssdm  33816  nodenselem8  33821  nolt02olem  33824