Theorem nosgnn0 31936

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 7620	. . . 4 ⊢ 1𝑜 ≠ ∅
2	1	nesymi 2880	. . 3 ⊢ ¬ ∅ = 1𝑜
3		nsuceq0 5843	. . . . 5 ⊢ suc 1𝑜 ≠ ∅
4		necom 2876	. . . . . 6 ⊢ (suc 1𝑜 ≠ ∅ ↔ ∅ ≠ suc 1𝑜)
5		df-2o 7606	. . . . . . 7 ⊢ 2𝑜 = suc 1𝑜
6	5	neeq2i 2888	. . . . . 6 ⊢ (∅ ≠ 2𝑜 ↔ ∅ ≠ suc 1𝑜)
7	4, 6	bitr4i 267	. . . . 5 ⊢ (suc 1𝑜 ≠ ∅ ↔ ∅ ≠ 2𝑜)
8	3, 7	mpbi 220	. . . 4 ⊢ ∅ ≠ 2𝑜
9	8	neii 2825	. . 3 ⊢ ¬ ∅ = 2𝑜
10	2, 9	pm3.2ni 917	. 2 ⊢ ¬ (∅ = 1𝑜 ∨ ∅ = 2𝑜)
11		0ex 4823	. . 3 ⊢ ∅ ∈ V
12	11	elpr 4231	. 2 ⊢ (∅ ∈ {1𝑜, 2𝑜} ↔ (∅ = 1𝑜 ∨ ∅ = 2𝑜))
13	10, 12	mtbir 312	1 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜}

Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∅c0 3948  {cpr 4212  suc csuc 5763  1_𝑜c1o 7598  2_𝑜c2o 7599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213  df-suc 5767  df-1o 7605  df-2o 7606

This theorem is referenced by:  nosgnn0i  31937  sltres  31940  noseponlem  31942  sltso  31952  nosepssdm  31961  nodenselem8  31966  nolt02olem  31969