Theorem elzs2 28416

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 28415	. 2 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
2		eln0s 28378	. . . . . 6 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ No → (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s )))
4		eln0s 28378	. . . . . 6 ⊢ (( -us ‘𝑁) ∈ ℕ0s ↔ (( -us ‘𝑁) ∈ ℕs ∨ ( -us ‘𝑁) = 0s ))
5		neg0s 28043	. . . . . . . . 9 ⊢ ( -us ‘ 0s ) = 0s
6	5	eqeq2i 2753	. . . . . . . 8 ⊢ (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ ( -us ‘𝑁) = 0s )
7		0no 27826	. . . . . . . . 9 ⊢ 0s ∈ No
8		negs11 28066	. . . . . . . . 9 ⊢ ((𝑁 ∈ No ∧ 0s ∈ No ) → (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ 𝑁 = 0s ))
9	7, 8	mpan2 697	. . . . . . . 8 ⊢ (𝑁 ∈ No → (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ 𝑁 = 0s ))
10	6, 9	bitr3id 286	. . . . . . 7 ⊢ (𝑁 ∈ No → (( -us ‘𝑁) = 0s ↔ 𝑁 = 0s ))
11	10	orbi2d 921	. . . . . 6 ⊢ (𝑁 ∈ No → ((( -us ‘𝑁) ∈ ℕs ∨ ( -us ‘𝑁) = 0s ) ↔ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
12	4, 11	bitrid 284	. . . . 5 ⊢ (𝑁 ∈ No → (( -us ‘𝑁) ∈ ℕ0s ↔ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
13	3, 12	orbi12d 924	. . . 4 ⊢ (𝑁 ∈ No → ((𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))))
14		3orcoma 1098	. . . . 5 ⊢ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 = 0s ∨ 𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs))
15		3orass 1095	. . . . 5 ⊢ ((𝑁 = 0s ∨ 𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)))
16		orcom 876	. . . . . 6 ⊢ ((𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ ((𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ∨ 𝑁 = 0s ))
17		orordir 935	. . . . . 6 ⊢ (((𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ∨ 𝑁 = 0s ) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
18	16, 17	bitri 276	. . . . 5 ⊢ ((𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
19	14, 15, 18	3bitrri 299	. . . 4 ⊢ (((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )) ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs))
20	13, 19	bitr2di 289	. . 3 ⊢ (𝑁 ∈ No → ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
21	20	pm5.32i 579	. 2 ⊢ ((𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
22	1, 21	bitr4i 279	1 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs)))

Syntax hints:   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∨ w3o 1091   = wceq 1547   ∈ wcel 2119  ‘cfv 6492   No csur 27628   0_s c0s 27822   -u_s cnegs 28036  ℕ_0scn0s 28329  ℕ_scnns 28330  ℤ_sczs 28395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-n0s 28331  df-nns 28332  df-zs 28396

This theorem is referenced by:  elnnzs  28418  elznns  28419