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Theorem elnns2 28344
Description: A positive surreal integer is a non-negative surreal integer greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)
Assertion
Ref Expression
elnns2 (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))

Proof of Theorem elnns2
StepHypRef Expression
1 elnns 28343 . 2 (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s𝐴 ≠ 0s ))
2 nesym 2997 . . . . 5 (𝐴 ≠ 0s ↔ ¬ 0s = 𝐴)
3 n0sge0 28341 . . . . . . . 8 (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
4 0sno 27871 . . . . . . . . 9 0s No
5 n0sno 28328 . . . . . . . . 9 (𝐴 ∈ ℕ0s𝐴 No )
6 sleloe 27799 . . . . . . . . 9 (( 0s No 𝐴 No ) → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
74, 5, 6sylancr 587 . . . . . . . 8 (𝐴 ∈ ℕ0s → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
83, 7mpbid 232 . . . . . . 7 (𝐴 ∈ ℕ0s → ( 0s <s 𝐴 ∨ 0s = 𝐴))
98orcomd 872 . . . . . 6 (𝐴 ∈ ℕ0s → ( 0s = 𝐴 ∨ 0s <s 𝐴))
109ord 865 . . . . 5 (𝐴 ∈ ℕ0s → (¬ 0s = 𝐴 → 0s <s 𝐴))
112, 10biimtrid 242 . . . 4 (𝐴 ∈ ℕ0s → (𝐴 ≠ 0s → 0s <s 𝐴))
12 sgt0ne0 27879 . . . 4 ( 0s <s 𝐴𝐴 ≠ 0s )
1311, 12impbid1 225 . . 3 (𝐴 ∈ ℕ0s → (𝐴 ≠ 0s ↔ 0s <s 𝐴))
1413pm5.32i 574 . 2 ((𝐴 ∈ ℕ0s𝐴 ≠ 0s ) ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
151, 14bitri 275 1 (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143   No csur 27684   <s cslt 27685   ≤s csle 27789   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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec2 27982  df-adds 27993  df-n0s 28320  df-nns 28321
This theorem is referenced by:  nnaddscl  28349  nnmulscl  28350
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