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Theorem elnns 28343
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 28321 . . 3 s = (ℕ0s ∖ { 0s })
21eleq2i 2833 . 2 (𝐴 ∈ ℕs𝐴 ∈ (ℕ0s ∖ { 0s }))
3 eldifsn 4786 . 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 2108  wne 2940  cdif 3948  {csn 4626   0s c0s 27867  0scnn0s 28318  scnns 28319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-sn 4627  df-nns 28321
This theorem is referenced by:  elnns2  28344  nnsge1  28346  eln0s  28358  dfnns2  28362
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