Theorem elnnzs 28788

Description: Positive surreal integer property expressed in terms of integers. (Contributed by Scott Fenton, 25-Jul-2025.)

Assertion

Ref	Expression
elnnzs	⊢ (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℤs ∧ 0s <s 𝑁))

Proof of Theorem elnnzs

Step	Hyp	Ref	Expression
1		nnno 28711	. . . 4 ⊢ (𝑁 ∈ ℕs → 𝑁 ∈ No)
2		orc 881	. . . 4 ⊢ (𝑁 ∈ ℕs → (𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
3		nnsgt0 28726	. . . 4 ⊢ (𝑁 ∈ ℕs → 0s <s 𝑁)
4	1, 2, 3	jca31 524	. . 3 ⊢ (𝑁 ∈ ℕs → ((𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))) ∧ 0s <s 𝑁))
5		idd 25	. . . . . . 7 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → (𝑁 ∈ ℕs → 𝑁 ∈ ℕs))
6		negscl 28423	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ No → ( -s‘𝑁) ∈ No)
7	6	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → ( -s‘𝑁) ∈ No)
8		0no 28196	. . . . . . . . . . . . . 14 ⊢ 0s ∈ No
9		ltnegs 28432	. . . . . . . . . . . . . 14 ⊢ (( 0s ∈ No ∧ 𝑁 ∈ No) → ( 0s <s 𝑁 ↔ ( -s‘𝑁) <s ( -s‘ 0s )))
10	8, 9	mpan 703	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ No → ( 0s <s 𝑁 ↔ ( -s‘𝑁) <s ( -s‘ 0s )))
11		neg0s 28413	. . . . . . . . . . . . . 14 ⊢ ( -s‘ 0s ) = 0s
12	11	breq2i 5110	. . . . . . . . . . . . 13 ⊢ (( -s‘𝑁) <s ( -s‘ 0s ) ↔ ( -s‘𝑁) <s 0s )
13	10, 12	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ No → ( 0s <s 𝑁 ↔ ( -s‘𝑁) <s 0s ))
14	13	biimpa 482	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → ( -s‘𝑁) <s 0s )
15		ltsasym 28106	. . . . . . . . . . . 12 ⊢ ((( -s‘𝑁) ∈ No ∧ 0s ∈ No) → (( -s‘𝑁) <s 0s → ¬ 0s <s ( -s‘𝑁)))
16	8, 15	mpan2 704	. . . . . . . . . . 11 ⊢ (( -s‘𝑁) ∈ No → (( -s‘𝑁) <s 0s → ¬ 0s <s ( -s‘𝑁)))
17	7, 14, 16	sylc 66	. . . . . . . . . 10 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → ¬ 0s <s ( -s‘𝑁))
18		nnsgt0 28726	. . . . . . . . . 10 ⊢ (( -s‘𝑁) ∈ ℕs → 0s <s ( -s‘𝑁))
19	17, 18	nsyl 141	. . . . . . . . 9 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → ¬ ( -s‘𝑁) ∈ ℕs)
20		gt0ne0s 28205	. . . . . . . . . . 11 ⊢ ( 0s <s 𝑁 → 𝑁 ≠ 0s )
21	20	adantl 487	. . . . . . . . . 10 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → 𝑁 ≠ 0s )
22	21	neneqd 2960	. . . . . . . . 9 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → ¬ 𝑁 = 0s )
23		ioran 999	. . . . . . . . 9 ⊢ (¬ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ) ↔ (¬ ( -s‘𝑁) ∈ ℕs ∧ ¬ 𝑁 = 0s ))
24	19, 22, 23	sylanbrc 595	. . . . . . . 8 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → ¬ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))
25	24	pm2.21d 122	. . . . . . 7 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → ((( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ) → 𝑁 ∈ ℕs))
26	5, 25	jaod 873	. . . . . 6 ⊢ ((𝑁 ∈ No ∧ 0s <s 𝑁) → ((𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )) → 𝑁 ∈ ℕs))
27	26	ex 418	. . . . 5 ⊢ (𝑁 ∈ No → ( 0s <s 𝑁 → ((𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )) → 𝑁 ∈ ℕs)))
28	27	com23 87	. . . 4 ⊢ (𝑁 ∈ No → ((𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )) → ( 0s <s 𝑁 → 𝑁 ∈ ℕs)))
29	28	imp31 423	. . 3 ⊢ (((𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))) ∧ 0s <s 𝑁) → 𝑁 ∈ ℕs)
30	4, 29	impbii 212	. 2 ⊢ (𝑁 ∈ ℕs ↔ ((𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))) ∧ 0s <s 𝑁))
31		elzs2 28786	. . . 4 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -s‘𝑁) ∈ ℕs)))
32		3orcomb 1110	. . . . . 6 ⊢ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -s‘𝑁) ∈ ℕs) ↔ (𝑁 ∈ ℕs ∨ ( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))
33		3orass 1106	. . . . . 6 ⊢ ((𝑁 ∈ ℕs ∨ ( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ) ↔ (𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
34	32, 33	bitri 278	. . . . 5 ⊢ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -s‘𝑁) ∈ ℕs) ↔ (𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
35	34	anbi2i 635	. . . 4 ⊢ ((𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -s‘𝑁) ∈ ℕs)) ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))))
36	31, 35	bitri 278	. . 3 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))))
37	36	anbi1i 636	. 2 ⊢ ((𝑁 ∈ ℤs ∧ 0s <s 𝑁) ↔ ((𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ (( -s‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))) ∧ 0s <s 𝑁))
38	30, 37	bitr4i 281	1 ⊢ (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℤs ∧ 0s <s 𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   class class class wbr 5102  ‘cfv 6538  Nocsur 27998   <s clts 27999   0_s c0s 28192   -_scnegs 28406  ℕ_scnns 28700  ℤ_sczs 28765

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

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 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-2o 8478  df-nadd 8676  df-no 28001  df-lts 28002  df-bday 28003  df-les 28103  df-slts 28145  df-cuts 28147  df-0s 28194  df-1s 28195  df-made 28214  df-old 28215  df-left 28217  df-right 28218  df-norec 28325  df-norec2 28336  df-adds 28347  df-negs 28408  df-subs 28409  df-n0s 28701  df-nns 28702  df-zs 28766

This theorem is used by: (None)