Theorem eln0s 28373

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 2939	. . . . 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 28358	. . 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 28349	. . . . 5 ⊢ 0s ∈ ℕ0s
13	11, 12	eqeltrdi 2847	. . . 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 1537   ∈ wcel 2106   ≠ wne 2938   0_s c0s 27882  ℕ_0scnn0s 28333  ℕ_scnns 28334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884  df-n0s 28335  df-nns 28336

This theorem is referenced by:  n0zs  28390  elzs2  28400  elznns  28403  expsp1  28427