Theorem nosgnn0i 31812

Description: If 𝑋 is a surreal sign, then it is not null. (Contributed by Scott Fenton, 3-Aug-2011.)

Hypothesis

Ref	Expression
nosgnn0i.1	⊢ 𝑋 ∈ {1𝑜, 2𝑜}

Assertion

Ref	Expression
nosgnn0i	⊢ ∅ ≠ 𝑋

Proof of Theorem nosgnn0i

Step	Hyp	Ref	Expression
1		nosgnn0 31811	. . 3 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1𝑜, 2𝑜}
3		eleq1 2689	. . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1𝑜, 2𝑜} ↔ 𝑋 ∈ {1𝑜, 2𝑜}))
4	2, 3	mpbiri 248	. . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1𝑜, 2𝑜})
5	1, 4	mto 188	. 2 ⊢ ¬ ∅ = 𝑋
6	5	neir 2797	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915  {cpr 4179  1_𝑜c1o 7553  2_𝑜c2o 7554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-suc 5729  df-1o 7560  df-2o 7561

This theorem is referenced by:  sltres  31815  noextenddif  31821  nolesgn2ores  31825  nosepnelem  31830  nosepdmlem  31833  nolt02o  31845  nosupbnd1lem3  31856  nosupbnd1lem5  31858  nosupbnd2lem1  31861