Theorem eln0s 28627

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 ∈ ℕs ∨ 𝐴 = 0s ))

Proof of Theorem eln0s

Step	Hyp	Ref	Expression
1		pm2.1 910	. . . 4 ⊢ (¬ 𝐴 = 0s ∨ 𝐴 = 0s )
2		df-ne 2958	. . . . 5 ⊢ (𝐴 ≠ 0s ↔ ¬ 𝐴 = 0s )
3	2	orbi1i 927	. . . 4 ⊢ ((𝐴 ≠ 0s ∨ 𝐴 = 0s ) ↔ (¬ 𝐴 = 0s ∨ 𝐴 = 0s ))
4	1, 3	mpbir 234	. . 3 ⊢ (𝐴 ≠ 0s ∨ 𝐴 = 0s )
5		ordir 1024	. . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ) ∧ (𝐴 ≠ 0s ∨ 𝐴 = 0s )))
6	4, 5	mpbiran2 723	. 2 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ))
7		elnns 28606	. . 3 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
8	7	orbi1i 927	. 2 ⊢ ((𝐴 ∈ ℕs ∨ 𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ))
9		orc 881	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ))
10		id 23	. . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s)
11		id 23	. . . . 5 ⊢ (𝐴 = 0s → 𝐴 = 0s )
12		0n0s 28595	. . . . 5 ⊢ 0s ∈ ℕ0s
13	11, 12	eqeltrdi 2870	. . . 4 ⊢ (𝐴 = 0s → 𝐴 ∈ ℕ0s)
14	10, 13	jaoi 871	. . 3 ⊢ ((𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ) → 𝐴 ∈ ℕ0s)
15	9, 14	impbii 212	. 2 ⊢ (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ))
16	6, 8, 15	3bitr4ri 307	1 ⊢ (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕs ∨ 𝐴 = 0s ))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   0_s c0s 28071  ℕ_0scn0s 28578  ℕ_scnns 28579

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-no 27880  df-lts 27881  df-bday 27882  df-slts 28024  df-cuts 28026  df-0s 28073  df-n0s 28580  df-nns 28581

This theorem is used by:  nnm1n0s  28641  n0zs  28655  elzs2  28665  elznns  28668  expsp1  28695