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Theorem elnnzs 28387
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
StepHypRef Expression
1 nnsno 28329 . . . 4 (𝑁 ∈ ℕs𝑁 No )
2 orc 868 . . . 4 (𝑁 ∈ ℕs → (𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s )))
3 nnsgt0 28342 . . . 4 (𝑁 ∈ ℕs → 0s <s 𝑁)
41, 2, 3jca31 514 . . 3 (𝑁 ∈ ℕs → ((𝑁 No ∧ (𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s ))) ∧ 0s <s 𝑁))
5 idd 24 . . . . . . 7 ((𝑁 No ∧ 0s <s 𝑁) → (𝑁 ∈ ℕs𝑁 ∈ ℕs))
6 negscl 28068 . . . . . . . . . . . 12 (𝑁 No → ( -us𝑁) ∈ No )
76adantr 480 . . . . . . . . . . 11 ((𝑁 No ∧ 0s <s 𝑁) → ( -us𝑁) ∈ No )
8 0sno 27871 . . . . . . . . . . . . . 14 0s No
9 sltneg 28077 . . . . . . . . . . . . . 14 (( 0s No 𝑁 No ) → ( 0s <s 𝑁 ↔ ( -us𝑁) <s ( -us ‘ 0s )))
108, 9mpan 690 . . . . . . . . . . . . 13 (𝑁 No → ( 0s <s 𝑁 ↔ ( -us𝑁) <s ( -us ‘ 0s )))
11 negs0s 28058 . . . . . . . . . . . . . 14 ( -us ‘ 0s ) = 0s
1211breq2i 5151 . . . . . . . . . . . . 13 (( -us𝑁) <s ( -us ‘ 0s ) ↔ ( -us𝑁) <s 0s )
1310, 12bitrdi 287 . . . . . . . . . . . 12 (𝑁 No → ( 0s <s 𝑁 ↔ ( -us𝑁) <s 0s ))
1413biimpa 476 . . . . . . . . . . 11 ((𝑁 No ∧ 0s <s 𝑁) → ( -us𝑁) <s 0s )
15 sltasym 27793 . . . . . . . . . . . 12 ((( -us𝑁) ∈ No ∧ 0s No ) → (( -us𝑁) <s 0s → ¬ 0s <s ( -us𝑁)))
168, 15mpan2 691 . . . . . . . . . . 11 (( -us𝑁) ∈ No → (( -us𝑁) <s 0s → ¬ 0s <s ( -us𝑁)))
177, 14, 16sylc 65 . . . . . . . . . 10 ((𝑁 No ∧ 0s <s 𝑁) → ¬ 0s <s ( -us𝑁))
18 nnsgt0 28342 . . . . . . . . . 10 (( -us𝑁) ∈ ℕs → 0s <s ( -us𝑁))
1917, 18nsyl 140 . . . . . . . . 9 ((𝑁 No ∧ 0s <s 𝑁) → ¬ ( -us𝑁) ∈ ℕs)
20 sgt0ne0 27879 . . . . . . . . . . 11 ( 0s <s 𝑁𝑁 ≠ 0s )
2120adantl 481 . . . . . . . . . 10 ((𝑁 No ∧ 0s <s 𝑁) → 𝑁 ≠ 0s )
2221neneqd 2945 . . . . . . . . 9 ((𝑁 No ∧ 0s <s 𝑁) → ¬ 𝑁 = 0s )
23 ioran 986 . . . . . . . . 9 (¬ (( -us𝑁) ∈ ℕs𝑁 = 0s ) ↔ (¬ ( -us𝑁) ∈ ℕs ∧ ¬ 𝑁 = 0s ))
2419, 22, 23sylanbrc 583 . . . . . . . 8 ((𝑁 No ∧ 0s <s 𝑁) → ¬ (( -us𝑁) ∈ ℕs𝑁 = 0s ))
2524pm2.21d 121 . . . . . . 7 ((𝑁 No ∧ 0s <s 𝑁) → ((( -us𝑁) ∈ ℕs𝑁 = 0s ) → 𝑁 ∈ ℕs))
265, 25jaod 860 . . . . . 6 ((𝑁 No ∧ 0s <s 𝑁) → ((𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s )) → 𝑁 ∈ ℕs))
2726ex 412 . . . . 5 (𝑁 No → ( 0s <s 𝑁 → ((𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s )) → 𝑁 ∈ ℕs)))
2827com23 86 . . . 4 (𝑁 No → ((𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s )) → ( 0s <s 𝑁𝑁 ∈ ℕs)))
2928imp31 417 . . 3 (((𝑁 No ∧ (𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s ))) ∧ 0s <s 𝑁) → 𝑁 ∈ ℕs)
304, 29impbii 209 . 2 (𝑁 ∈ ℕs ↔ ((𝑁 No ∧ (𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s ))) ∧ 0s <s 𝑁))
31 elzs2 28385 . . . 4 (𝑁 ∈ ℤs ↔ (𝑁 No ∧ (𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs)))
32 3orcomb 1094 . . . . . 6 ((𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs) ↔ (𝑁 ∈ ℕs ∨ ( -us𝑁) ∈ ℕs𝑁 = 0s ))
33 3orass 1090 . . . . . 6 ((𝑁 ∈ ℕs ∨ ( -us𝑁) ∈ ℕs𝑁 = 0s ) ↔ (𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s )))
3432, 33bitri 275 . . . . 5 ((𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs) ↔ (𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s )))
3534anbi2i 623 . . . 4 ((𝑁 No ∧ (𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs)) ↔ (𝑁 No ∧ (𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s ))))
3631, 35bitri 275 . . 3 (𝑁 ∈ ℤs ↔ (𝑁 No ∧ (𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s ))))
3736anbi1i 624 . 2 ((𝑁 ∈ ℤs ∧ 0s <s 𝑁) ↔ ((𝑁 No ∧ (𝑁 ∈ ℕs ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s ))) ∧ 0s <s 𝑁))
3830, 37bitr4i 278 1 (𝑁 ∈ ℕs ↔ (𝑁 ∈ ℤs ∧ 0s <s 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3o 1086   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  cfv 6561   No csur 27684   <s cslt 27685   0s c0s 27867   -us cnegs 28051  scnns 28319  sczs 28364
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-n0s 28320  df-nns 28321  df-zs 28365
This theorem is referenced by: (None)
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