Theorem nosgnn0 27578

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 8502	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2993	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6446	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 2989	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8481	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3001	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 278	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 229	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2937	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 879	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5301	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4647	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 323	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 846   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  ∅c0 4318  {cpr 4626  suc csuc 6365  1_oc1o 8473  2_oc2o 8474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-v 3471  df-dif 3947  df-un 3949  df-nul 4319  df-sn 4625  df-pr 4627  df-suc 6369  df-1o 8480  df-2o 8481

This theorem is referenced by:  nosgnn0i  27579  sltres  27582  noseponlem  27584  sltso  27596  nosepssdm  27606  nodenselem8  27611  nolt02olem  27614