Theorem nosgnn0 31055

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 7462	. . . 4 ⊢ 1𝑜 ≠ ∅
2	1	nesymi 2839	. . 3 ⊢ ¬ ∅ = 1𝑜
3		nsuceq0 5722	. . . . 5 ⊢ suc 1𝑜 ≠ ∅
4		necom 2835	. . . . . 6 ⊢ (suc 1𝑜 ≠ ∅ ↔ ∅ ≠ suc 1𝑜)
5		df-2o 7448	. . . . . . 7 ⊢ 2𝑜 = suc 1𝑜
6	5	neeq2i 2847	. . . . . 6 ⊢ (∅ ≠ 2𝑜 ↔ ∅ ≠ suc 1𝑜)
7	4, 6	bitr4i 266	. . . . 5 ⊢ (suc 1𝑜 ≠ ∅ ↔ ∅ ≠ 2𝑜)
8	3, 7	mpbi 219	. . . 4 ⊢ ∅ ≠ 2𝑜
9	8	neii 2784	. . 3 ⊢ ¬ ∅ = 2𝑜
10	2, 9	pm3.2ni 895	. 2 ⊢ ¬ (∅ = 1𝑜 ∨ ∅ = 2𝑜)
11		0ex 4718	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4146	. 2 ⊢ (∅ ∈ {1𝑜, 2𝑜} ↔ (∅ = 1𝑜 ∨ ∅ = 2𝑜))
13	10, 12	mtbir 312	1 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜}

Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  {cpr 4127  suc csuc 5642  1_𝑜c1o 7440  2_𝑜c2o 7441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-suc 5646  df-1o 7447  df-2o 7448

This theorem is referenced by:  nosgnn0i  31056  sltres  31061  noseponlem  31065  sltso  31068  nodenselem3  31082  nodenselem8  31087