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Theorem elzs2 28386
Description: A surreal integer is either a positive integer, zero, or the negative of a positive integer. (Contributed by Scott Fenton, 25-Jul-2025.)
Assertion
Ref Expression
elzs2 (𝑁 ∈ ℤs ↔ (𝑁 No ∧ (𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs)))

Proof of Theorem elzs2
StepHypRef Expression
1 elzn0s 28385 . 2 (𝑁 ∈ ℤs ↔ (𝑁 No ∧ (𝑁 ∈ ℕ0s ∨ ( -us𝑁) ∈ ℕ0s)))
2 eln0s 28359 . . . . . 6 (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs𝑁 = 0s ))
32a1i 11 . . . . 5 (𝑁 No → (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs𝑁 = 0s )))
4 eln0s 28359 . . . . . 6 (( -us𝑁) ∈ ℕ0s ↔ (( -us𝑁) ∈ ℕs ∨ ( -us𝑁) = 0s ))
5 negs0s 28059 . . . . . . . . 9 ( -us ‘ 0s ) = 0s
65eqeq2i 2749 . . . . . . . 8 (( -us𝑁) = ( -us ‘ 0s ) ↔ ( -us𝑁) = 0s )
7 0sno 27872 . . . . . . . . 9 0s No
8 negs11 28082 . . . . . . . . 9 ((𝑁 No ∧ 0s No ) → (( -us𝑁) = ( -us ‘ 0s ) ↔ 𝑁 = 0s ))
97, 8mpan2 691 . . . . . . . 8 (𝑁 No → (( -us𝑁) = ( -us ‘ 0s ) ↔ 𝑁 = 0s ))
106, 9bitr3id 285 . . . . . . 7 (𝑁 No → (( -us𝑁) = 0s𝑁 = 0s ))
1110orbi2d 915 . . . . . 6 (𝑁 No → ((( -us𝑁) ∈ ℕs ∨ ( -us𝑁) = 0s ) ↔ (( -us𝑁) ∈ ℕs𝑁 = 0s )))
124, 11bitrid 283 . . . . 5 (𝑁 No → (( -us𝑁) ∈ ℕ0s ↔ (( -us𝑁) ∈ ℕs𝑁 = 0s )))
133, 12orbi12d 918 . . . 4 (𝑁 No → ((𝑁 ∈ ℕ0s ∨ ( -us𝑁) ∈ ℕ0s) ↔ ((𝑁 ∈ ℕs𝑁 = 0s ) ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s ))))
14 3orcoma 1092 . . . . 5 ((𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs) ↔ (𝑁 = 0s𝑁 ∈ ℕs ∨ ( -us𝑁) ∈ ℕs))
15 3orass 1089 . . . . 5 ((𝑁 = 0s𝑁 ∈ ℕs ∨ ( -us𝑁) ∈ ℕs) ↔ (𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us𝑁) ∈ ℕs)))
16 orcom 870 . . . . . 6 ((𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us𝑁) ∈ ℕs)) ↔ ((𝑁 ∈ ℕs ∨ ( -us𝑁) ∈ ℕs) ∨ 𝑁 = 0s ))
17 orordir 929 . . . . . 6 (((𝑁 ∈ ℕs ∨ ( -us𝑁) ∈ ℕs) ∨ 𝑁 = 0s ) ↔ ((𝑁 ∈ ℕs𝑁 = 0s ) ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s )))
1816, 17bitri 275 . . . . 5 ((𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us𝑁) ∈ ℕs)) ↔ ((𝑁 ∈ ℕs𝑁 = 0s ) ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s )))
1914, 15, 183bitrri 298 . . . 4 (((𝑁 ∈ ℕs𝑁 = 0s ) ∨ (( -us𝑁) ∈ ℕs𝑁 = 0s )) ↔ (𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs))
2013, 19bitr2di 288 . . 3 (𝑁 No → ((𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs) ↔ (𝑁 ∈ ℕ0s ∨ ( -us𝑁) ∈ ℕ0s)))
2120pm5.32i 574 . 2 ((𝑁 No ∧ (𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs)) ↔ (𝑁 No ∧ (𝑁 ∈ ℕ0s ∨ ( -us𝑁) ∈ ℕ0s)))
221, 21bitr4i 278 1 (𝑁 ∈ ℤs ↔ (𝑁 No ∧ (𝑁 ∈ ℕs𝑁 = 0s ∨ ( -us𝑁) ∈ ℕs)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wo 847  w3o 1085   = wceq 1539  wcel 2107  cfv 6560   No csur 27685   0s c0s 27868   -us cnegs 28052  0scnn0s 28319  scnns 28320  sczs 28365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-1s 27871  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054  df-subs 28055  df-n0s 28321  df-nns 28322  df-zs 28366
This theorem is referenced by:  elnnzs  28388  elznns  28389
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