Theorem elnns2 28285

Description: A positive surreal integer is a non-negative surreal integer greater than zero. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
elnns2	⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))

Proof of Theorem elnns2

Step	Hyp	Ref	Expression
1		elnns 28284	. 2 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
2		nesym 2988	. . . . 5 ⊢ (𝐴 ≠ 0s ↔ ¬ 0s = 𝐴)
3		n0sge0 28282	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
4		0sno 27790	. . . . . . . . 9 ⊢ 0s ∈ No
5		n0sno 28268	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
6		sleloe 27718	. . . . . . . . 9 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No ) → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
7	4, 5, 6	sylancr 587	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
8	3, 7	mpbid 232	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → ( 0s <s 𝐴 ∨ 0s = 𝐴))
9	8	orcomd 871	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ( 0s = 𝐴 ∨ 0s <s 𝐴))
10	9	ord 864	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → (¬ 0s = 𝐴 → 0s <s 𝐴))
11	2, 10	biimtrid 242	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≠ 0s → 0s <s 𝐴))
12		sgt0ne0 27799	. . . 4 ⊢ ( 0s <s 𝐴 → 𝐴 ≠ 0s )
13	11, 12	impbid1 225	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≠ 0s ↔ 0s <s 𝐴))
14	13	pm5.32i 574	. 2 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
15	1, 14	bitri 275	1 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108   ≠ wne 2932   class class class wbr 5119   No csur 27603   <s cslt 27604   ≤s csle 27708   0_s c0s 27786  ℕ_0scnn0s 28258  ℕ_scnns 28259

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-nadd 8678  df-no 27606  df-slt 27607  df-bday 27608  df-sle 27709  df-sslt 27745  df-scut 27747  df-0s 27788  df-1s 27789  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-norec2 27908  df-adds 27919  df-n0s 28260  df-nns 28261

This theorem is referenced by:  nnaddscl  28290  nnmulscl  28291