Theorem nosgnn0 27780

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 8460	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3017	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6435	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 3013	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8442	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3025	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 281	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 233	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 2962	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 893	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5262	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4610	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 326	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 860   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∅c0 4288  {cpr 4587  suc csuc 6352  1_oc1o 8434  2_oc2o 8435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-un 3912  df-nul 4289  df-sn 4586  df-pr 4588  df-suc 6356  df-1o 8441  df-2o 8442

This theorem is referenced by:  nosgnn0i  27781  ltsres  27784  noseponlem  27786  ltsso  27798  nosepssdm  27808  nodenselem8  27813  nolt02olem  27816