Theorem nosgnn0 32345

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 7847	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3056	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6047	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 3052	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 7832	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3064	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 270	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 222	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 3001	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 909	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5016	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4422	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 315	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 878   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∅c0 4146  {cpr 4401  suc csuc 5969  1_oc1o 7824  2_oc2o 7825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-v 3416  df-dif 3801  df-un 3803  df-nul 4147  df-sn 4400  df-pr 4402  df-suc 5973  df-1o 7831  df-2o 7832

This theorem is referenced by:  nosgnn0i  32346  sltres  32349  noseponlem  32351  sltso  32361  nosepssdm  32370  nodenselem8  32375  nolt02olem  32378