Theorem nosgnn0 33861

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 8318	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3001	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6346	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 2997	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8298	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3009	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 277	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 229	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2945	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 878	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5231	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4584	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 323	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  {cpr 4563  suc csuc 6268  1_oc1o 8290  2_oc2o 8291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564  df-suc 6272  df-1o 8297  df-2o 8298

This theorem is referenced by:  nosgnn0i  33862  sltres  33865  noseponlem  33867  sltso  33879  nosepssdm  33889  nodenselem8  33894  nolt02olem  33897