Theorem nosgnn0 33158

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 8111	. . . 4 ⊢ 1o ≠ ∅
2	1	nesymi 3071	. . 3 ⊢ ¬ ∅ = 1o
3		nsuceq0 6264	. . . . 5 ⊢ suc 1o ≠ ∅
4		necom 3067	. . . . . 6 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ suc 1o)
5		df-2o 8095	. . . . . . 7 ⊢ 2o = suc 1o
6	5	neeq2i 3079	. . . . . 6 ⊢ (∅ ≠ 2o ↔ ∅ ≠ suc 1o)
7	4, 6	bitr4i 280	. . . . 5 ⊢ (suc 1o ≠ ∅ ↔ ∅ ≠ 2o)
8	3, 7	mpbi 232	. . . 4 ⊢ ∅ ≠ 2o
9	8	neii 3016	. . 3 ⊢ ¬ ∅ = 2o
10	2, 9	pm3.2ni 877	. 2 ⊢ ¬ (∅ = 1o ∨ ∅ = 2o)
11		0ex 5202	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4582	. 2 ⊢ (∅ ∈ {1o, 2o} ↔ (∅ = 1o ∨ ∅ = 2o))
13	10, 12	mtbir 325	1 ⊢ ¬ ∅ ∈ {1o, 2o}

Syntax hints:  ¬ wn 3   ∨ wo 843   = wceq 1531   ∈ wcel 2108   ≠ wne 3014  ∅c0 4289  {cpr 4561  suc csuc 6186  1_oc1o 8087  2_oc2o 8088

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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-nul 5201

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-v 3495  df-dif 3937  df-un 3939  df-nul 4290  df-sn 4560  df-pr 4562  df-suc 6190  df-1o 8094  df-2o 8095

This theorem is referenced by:  nosgnn0i  33159  sltres  33162  noseponlem  33164  sltso  33174  nosepssdm  33183  nodenselem8  33188  nolt02olem  33191