Theorem eln0s 28293

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 2929	. . . . 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 28274	. . 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 28264	. . . . 5 ⊢ 0s ∈ ℕ0s
13	11, 12	eqeltrdi 2839	. . . 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 1541   ∈ wcel 2111   ≠ wne 2928   0_s c0s 27772  ℕ_0scnn0s 28248  ℕ_scnns 28249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-no 27587  df-slt 27588  df-bday 27589  df-sslt 27727  df-scut 27729  df-0s 27774  df-n0s 28250  df-nns 28251

This theorem is referenced by:  nnm1n0s  28306  n0zs  28319  elzs2  28329  elznns  28332  expsp1  28358