Theorem nosgnn0 33278

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 8102	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3044	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6239	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 3040	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8086	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3052	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 281	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 233	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2989	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 878	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5175	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4548	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 326	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∅c0 4243  {cpr 4527  suc csuc 6161  1_oc1o 8078  2_oc2o 8079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528  df-suc 6165  df-1o 8085  df-2o 8086

This theorem is referenced by:  nosgnn0i  33279  sltres  33282  noseponlem  33284  sltso  33294  nosepssdm  33303  nodenselem8  33308  nolt02olem  33311