Theorem nosgnn0 27703

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 8526	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2998	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6467	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 2994	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8507	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3006	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 278	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 230	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2942	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 881	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5307	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4650	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 323	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 848   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∅c0 4333  {cpr 4628  suc csuc 6386  1_oc1o 8499  2_oc2o 8500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-pr 4629  df-suc 6390  df-1o 8506  df-2o 8507

This theorem is referenced by:  nosgnn0i  27704  sltres  27707  noseponlem  27709  sltso  27721  nosepssdm  27731  nodenselem8  27736  nolt02olem  27739