Theorem nosgnn0 33494

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 8143	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 2991	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6246	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 2987	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8125	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 2999	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 281	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 233	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2936	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 880	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5172	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4536	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 326	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 846   = wceq 1542   ∈ wcel 2113   ≠ wne 2934  ∅c0 4209  {cpr 4515  suc csuc 6168  1_oc1o 8117  2_oc2o 8118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710  ax-nul 5171

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3399  df-dif 3844  df-un 3846  df-nul 4210  df-sn 4514  df-pr 4516  df-suc 6172  df-1o 8124  df-2o 8125

This theorem is referenced by:  nosgnn0i  33495  sltres  33498  noseponlem  33500  sltso  33512  nosepssdm  33522  nodenselem8  33527  nolt02olem  33530