Theorem eln0s 28258

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		pm2.1 896	. . . 4 ⊢ (¬ 𝐴 = 0s ∨ 𝐴 = 0s )
2		df-ne 2927	. . . . 5 ⊢ (𝐴 ≠ 0s ↔ ¬ 𝐴 = 0s )
3	2	orbi1i 913	. . . 4 ⊢ ((𝐴 ≠ 0s ∨ 𝐴 = 0s ) ↔ (¬ 𝐴 = 0s ∨ 𝐴 = 0s ))
4	1, 3	mpbir 231	. . 3 ⊢ (𝐴 ≠ 0s ∨ 𝐴 = 0s )
5		ordir 1008	. . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ) ∧ (𝐴 ≠ 0s ∨ 𝐴 = 0s )))
6	4, 5	mpbiran2 710	. 2 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ))
7		elnns 28239	. . 3 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
8	7	orbi1i 913	. 2 ⊢ ((𝐴 ∈ ℕs ∨ 𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ))
9		orc 867	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ))
10		id 22	. . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s)
11		id 22	. . . . 5 ⊢ (𝐴 = 0s → 𝐴 = 0s )
12		0n0s 28229	. . . . 5 ⊢ 0s ∈ ℕ0s
13	11, 12	eqeltrdi 2837	. . . 4 ⊢ (𝐴 = 0s → 𝐴 ∈ ℕ0s)
14	10, 13	jaoi 857	. . 3 ⊢ ((𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ) → 𝐴 ∈ ℕ0s)
15	9, 14	impbii 209	. 2 ⊢ (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ))
16	6, 8, 15	3bitr4ri 304	1 ⊢ (𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕs ∨ 𝐴 = 0s ))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   0_s c0s 27741  ℕ_0scnn0s 28213  ℕ_scnns 28214

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-0s 27743  df-n0s 28215  df-nns 28216

This theorem is referenced by:  nnm1n0s  28271  n0zs  28284  elzs2  28294  elznns  28297  expsp1  28322