Theorem elzs2 28603

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 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs)))

Proof of Theorem elzs2

Step	Hyp	Ref	Expression
1		elzn0s 28602	. 2 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
2		eln0s 28565	. . . . . 6 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ No → (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s )))
4		eln0s 28565	. . . . . 6 ⊢ (( -us ‘𝑁) ∈ ℕ0s ↔ (( -us ‘𝑁) ∈ ℕs ∨ ( -us ‘𝑁) = 0s ))
5		neg0s 28230	. . . . . . . . 9 ⊢ ( -us ‘ 0s ) = 0s
6	5	eqeq2i 2775	. . . . . . . 8 ⊢ (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ ( -us ‘𝑁) = 0s )
7		0no 28013	. . . . . . . . 9 ⊢ 0s ∈ No
8		negs11 28253	. . . . . . . . 9 ⊢ ((𝑁 ∈ No ∧ 0s ∈ No ) → (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ 𝑁 = 0s ))
9	7, 8	mpan2 703	. . . . . . . 8 ⊢ (𝑁 ∈ No → (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ 𝑁 = 0s ))
10	6, 9	bitr3id 288	. . . . . . 7 ⊢ (𝑁 ∈ No → (( -us ‘𝑁) = 0s ↔ 𝑁 = 0s ))
11	10	orbi2d 928	. . . . . 6 ⊢ (𝑁 ∈ No → ((( -us ‘𝑁) ∈ ℕs ∨ ( -us ‘𝑁) = 0s ) ↔ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
12	4, 11	bitrid 286	. . . . 5 ⊢ (𝑁 ∈ No → (( -us ‘𝑁) ∈ ℕ0s ↔ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
13	3, 12	orbi12d 931	. . . 4 ⊢ (𝑁 ∈ No → ((𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))))
14		3orcoma 1108	. . . . 5 ⊢ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 = 0s ∨ 𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs))
15		3orass 1105	. . . . 5 ⊢ ((𝑁 = 0s ∨ 𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)))
16		orcom 883	. . . . . 6 ⊢ ((𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ ((𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ∨ 𝑁 = 0s ))
17		orordir 942	. . . . . 6 ⊢ (((𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ∨ 𝑁 = 0s ) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
18	16, 17	bitri 278	. . . . 5 ⊢ ((𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
19	14, 15, 18	3bitrri 301	. . . 4 ⊢ (((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )) ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs))
20	13, 19	bitr2di 291	. . 3 ⊢ (𝑁 ∈ No → ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
21	20	pm5.32i 584	. 2 ⊢ ((𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
22	1, 21	bitr4i 281	1 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1101   = wceq 1569   ∈ wcel 2142  ‘cfv 6536   No csur 27815   0_s c0s 28009   -u_s cnegs 28223  ℕ_0scn0s 28516  ℕ_scnns 28517  ℤ_sczs 28582

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-n0s 28518  df-nns 28519  df-zs 28583

This theorem is used by:  elnnzs  28605  elznns  28606