Theorem elzs2 28719

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 28718	. 2 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
2		eln0s 28681	. . . . . 6 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ No → (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s )))
4		eln0s 28681	. . . . . 6 ⊢ (( -us ‘𝑁) ∈ ℕ0s ↔ (( -us ‘𝑁) ∈ ℕs ∨ ( -us ‘𝑁) = 0s ))
5		neg0s 28346	. . . . . . . . 9 ⊢ ( -us ‘ 0s ) = 0s
6	5	eqeq2i 2773	. . . . . . . 8 ⊢ (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ ( -us ‘𝑁) = 0s )
7		0no 28129	. . . . . . . . 9 ⊢ 0s ∈ No
8		negs11 28369	. . . . . . . . 9 ⊢ ((𝑁 ∈ No ∧ 0s ∈ No ) → (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ 𝑁 = 0s ))
9	7, 8	mpan2 704	. . . . . . . 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 929	. . . . . 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 932	. . . 4 ⊢ (𝑁 ∈ No → ((𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))))
14		3orcoma 1109	. . . . 5 ⊢ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 = 0s ∨ 𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs))
15		3orass 1106	. . . . 5 ⊢ ((𝑁 = 0s ∨ 𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)))
16		orcom 884	. . . . . 6 ⊢ ((𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ ((𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ∨ 𝑁 = 0s ))
17		orordir 943	. . . . . 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 585	. 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 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  ‘cfv 6527   No csur 27931   0_s c0s 28125   -u_s cnegs 28339  ℕ_0scn0s 28632  ℕ_scnns 28633  ℤ_sczs 28698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  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 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27934  df-lts 27935  df-bday 27936  df-les 28036  df-slts 28078  df-cuts 28080  df-0s 28127  df-1s 28128  df-made 28147  df-old 28148  df-left 28150  df-right 28151  df-norec 28258  df-norec2 28269  df-adds 28280  df-negs 28341  df-subs 28342  df-n0s 28634  df-nns 28635  df-zs 28699

This theorem is used by:  elnnzs  28721  elznns  28722