Theorem nosgnn0i 31937

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 31936	. . 3 ⊢ ¬ ∅ ∈ {1𝑜, 2𝑜}
2		nosgnn0i.1	. . . 4 ⊢ 𝑋 ∈ {1𝑜, 2𝑜}
3		eleq1 2718	. . . 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 2826	1 ⊢ ∅ ≠ 𝑋

Syntax hints:   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∅c0 3948  {cpr 4212  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:  sltres  31940  noextenddif  31946  nolesgn2ores  31950  nosepnelem  31955  nosepdmlem  31958  nolt02o  31970  nosupbnd1lem3  31981  nosupbnd1lem5  31983  nosupbnd2lem1  31986