Theorem eln0s 28530

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		pm2.1 909	. . . 4 ⊢ (¬ 𝐴 = 0s ∨ 𝐴 = 0s )
2		df-ne 2957	. . . . 5 ⊢ (𝐴 ≠ 0s ↔ ¬ 𝐴 = 0s )
3	2	orbi1i 926	. . . 4 ⊢ ((𝐴 ≠ 0s ∨ 𝐴 = 0s ) ↔ (¬ 𝐴 = 0s ∨ 𝐴 = 0s ))
4	1, 3	mpbir 234	. . 3 ⊢ (𝐴 ≠ 0s ∨ 𝐴 = 0s )
5		ordir 1022	. . 3 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ) ∧ (𝐴 ≠ 0s ∨ 𝐴 = 0s )))
6	4, 5	mpbiran2 722	. 2 ⊢ (((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ) ↔ (𝐴 ∈ ℕ0s ∨ 𝐴 = 0s ))
7		elnns 28509	. . 3 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
8	7	orbi1i 926	. 2 ⊢ ((𝐴 ∈ ℕs ∨ 𝐴 = 0s ) ↔ ((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ∨ 𝐴 = 0s ))
9		orc 880	. . 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 28498	. . . . 5 ⊢ 0s ∈ ℕ0s
13	11, 12	eqeltrdi 2869	. . . 4 ⊢ (𝐴 = 0s → 𝐴 ∈ ℕ0s)
14	10, 13	jaoi 870	. . 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 ))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1568   ∈ wcel 2141   ≠ wne 2956   0_s c0s 27974  ℕ_0scn0s 28481  ℕ_scnns 28482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-no 27783  df-lts 27784  df-bday 27785  df-slts 27927  df-cuts 27929  df-0s 27976  df-n0s 28483  df-nns 28484

This theorem is referenced by:  nnm1n0s  28544  n0zs  28558  elzs2  28568  elznns  28571  expsp1  28598