Theorem nosgnn0i 33161

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 33160	. . 3 ⊢ ¬ ∅ ∈ {1o, 2o}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1o, 2o}
3		eleq1 2900	. . . 4 ⊢ (∅ = 𝑋 → (∅ ∈ {1o, 2o} ↔ 𝑋 ∈ {1o, 2o}))
4	2, 3	mpbiri 260	. . 3 ⊢ (∅ = 𝑋 → ∅ ∈ {1o, 2o})
5	1, 4	mto 199	. 2 ⊢ ¬ ∅ = 𝑋
6	5	neir 3019	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∅c0 4290  {cpr 4562  1_oc1o 8089  2_oc2o 8090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4561  df-pr 4563  df-suc 6191  df-1o 8096  df-2o 8097

This theorem is referenced by:  sltres  33164  noextenddif  33170  nolesgn2ores  33174  nosepnelem  33179  nosepdmlem  33182  nolt02o  33194  nosupbnd1lem3  33205  nosupbnd1lem5  33207  nosupbnd2lem1  33210