Theorem nosgnn0 27721

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 8544	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3004	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6478	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 3000	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8523	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3012	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 278	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 230	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2948	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 879	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5325	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4672	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 323	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 846   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∅c0 4352  {cpr 4650  suc csuc 6397  1_oc1o 8515  2_oc2o 8516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-pr 4651  df-suc 6401  df-1o 8522  df-2o 8523

This theorem is referenced by:  nosgnn0i  27722  sltres  27725  noseponlem  27727  sltso  27739  nosepssdm  27749  nodenselem8  27754  nolt02olem  27757