Theorem elnns2 28739

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 28738	. 2 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ))
2		nesym 3014	. . . . 5 ⊢ (𝐴 ≠ 0s ↔ ¬ 0s = 𝐴)
3		n0sge0 28736	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)
4		0no 28207	. . . . . . . . 9 ⊢ 0s ∈ No
5		n0no 28721	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No)
6		lesloe 28123	. . . . . . . . 9 ⊢ (( 0s ∈ No ∧ 𝐴 ∈ No) → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
7	4, 5, 6	sylancr 599	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → ( 0s ≤s 𝐴 ↔ ( 0s <s 𝐴 ∨ 0s = 𝐴)))
8	3, 7	mpbid 235	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → ( 0s <s 𝐴 ∨ 0s = 𝐴))
9	8	orcomd 885	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ( 0s = 𝐴 ∨ 0s <s 𝐴))
10	9	ord 878	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → (¬ 0s = 𝐴 → 0s <s 𝐴))
11	2, 10	biimtrid 245	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≠ 0s → 0s <s 𝐴))
12		gt0ne0s 28216	. . . 4 ⊢ ( 0s <s 𝐴 → 𝐴 ≠ 0s )
13	11, 12	impbid1 228	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 ≠ 0s ↔ 0s <s 𝐴))
14	13	pm5.32i 585	. 2 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐴 ≠ 0s ) ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))
15	1, 14	bitri 278	1 ⊢ (𝐴 ∈ ℕs ↔ (𝐴 ∈ ℕ0s ∧ 0s <s 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2146   ≠ wne 2958   class class class wbr 5105  Nocsur 28009   <s clts 28010   ≤s cles 28113   0_s c0s 28203  ℕ_0scn0s 28710  ℕ_scnns 28711

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4838  df-ot 4861  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-se 5608  df-we 5609  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 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-2o 8488  df-nadd 8686  df-no 28012  df-lts 28013  df-bday 28014  df-les 28114  df-slts 28156  df-cuts 28158  df-0s 28205  df-1s 28206  df-made 28225  df-old 28226  df-left 28228  df-right 28229  df-norec2 28347  df-adds 28358  df-n0s 28712  df-nns 28713

This theorem is used by:  nnaddscl  28744  nnmulscl  28745