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Theorem eln0s 28358
Description: A non-negative surreal integer is zero or a positive surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
Assertion
Ref Expression
eln0s (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕs𝐴 = 0s ))

Proof of Theorem eln0s
StepHypRef Expression
1 pm2.1 897 . . . 4 𝐴 = 0s𝐴 = 0s )
2 df-ne 2941 . . . . 5 (𝐴 ≠ 0s ↔ ¬ 𝐴 = 0s )
32orbi1i 914 . . . 4 ((𝐴 ≠ 0s𝐴 = 0s ) ↔ (¬ 𝐴 = 0s𝐴 = 0s ))
41, 3mpbir 231 . . 3 (𝐴 ≠ 0s𝐴 = 0s )
5 ordir 1009 . . 3 (((𝐴 ∈ ℕ0s𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s𝐴 = 0s ) ∧ (𝐴 ≠ 0s𝐴 = 0s )))
64, 5mpbiran2 710 . 2 (((𝐴 ∈ ℕ0s𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ (𝐴 ∈ ℕ0s𝐴 = 0s ))
7 elnns 28343 . . 3 (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s𝐴 ≠ 0s ))
87orbi1i 914 . 2 ((𝐴 ∈ ℕs𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s𝐴 ≠ 0s ) ∨ 𝐴 = 0s ))
9 orc 868 . . 3 (𝐴 ∈ ℕ0s → (𝐴 ∈ ℕ0s𝐴 = 0s ))
10 id 22 . . . 4 (𝐴 ∈ ℕ0s𝐴 ∈ ℕ0s)
11 id 22 . . . . 5 (𝐴 = 0s𝐴 = 0s )
12 0n0s 28334 . . . . 5 0s ∈ ℕ0s
1311, 12eqeltrdi 2849 . . . 4 (𝐴 = 0s𝐴 ∈ ℕ0s)
1410, 13jaoi 858 . . 3 ((𝐴 ∈ ℕ0s𝐴 = 0s ) → 𝐴 ∈ ℕ0s)
159, 14impbii 209 . 2 (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕ0s𝐴 = 0s ))
166, 8, 153bitr4ri 304 1 (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕs𝐴 = 0s ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940   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-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-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-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869  df-n0s 28320  df-nns 28321
This theorem is referenced by:  n0zs  28375  elzs2  28385  elznns  28388  expsp1  28412
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