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Theorem elnns 28358
Description: Membership in the positive surreal integers. (Contributed by Scott Fenton, 15-Apr-2025.)
Assertion
Ref Expression
elnns (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s𝐴 ≠ 0s ))

Proof of Theorem elnns
StepHypRef Expression
1 df-nns 28336 . . 3 s = (ℕ0s ∖ { 0s })
21eleq2i 2831 . 2 (𝐴 ∈ ℕs𝐴 ∈ (ℕ0s ∖ { 0s }))
3 eldifsn 4791 . 2 (𝐴 ∈ (ℕ0s ∖ { 0s }) ↔ (𝐴 ∈ ℕ0s𝐴 ≠ 0s ))
42, 3bitri 275 1 (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s𝐴 ≠ 0s ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  wne 2938  cdif 3960  {csn 4631   0s c0s 27882  0scnn0s 28333  scnns 28334
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-sn 4632  df-nns 28336
This theorem is referenced by:  elnns2  28359  nnsge1  28361  eln0s  28373  dfnns2  28377
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