Theorem elzs2 28391

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 28390	. 2 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
2		eln0s 28353	. . . . . 6 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ No → (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s )))
4		eln0s 28353	. . . . . 6 ⊢ (( -us ‘𝑁) ∈ ℕ0s ↔ (( -us ‘𝑁) ∈ ℕs ∨ ( -us ‘𝑁) = 0s ))
5		neg0s 28018	. . . . . . . . 9 ⊢ ( -us ‘ 0s ) = 0s
6	5	eqeq2i 2750	. . . . . . . 8 ⊢ (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ ( -us ‘𝑁) = 0s )
7		0no 27801	. . . . . . . . 9 ⊢ 0s ∈ No
8		negs11 28041	. . . . . . . . 9 ⊢ ((𝑁 ∈ No ∧ 0s ∈ No ) → (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ 𝑁 = 0s ))
9	7, 8	mpan2 692	. . . . . . . 8 ⊢ (𝑁 ∈ No → (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ 𝑁 = 0s ))
10	6, 9	bitr3id 285	. . . . . . 7 ⊢ (𝑁 ∈ No → (( -us ‘𝑁) = 0s ↔ 𝑁 = 0s ))
11	10	orbi2d 916	. . . . . 6 ⊢ (𝑁 ∈ No → ((( -us ‘𝑁) ∈ ℕs ∨ ( -us ‘𝑁) = 0s ) ↔ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
12	4, 11	bitrid 283	. . . . 5 ⊢ (𝑁 ∈ No → (( -us ‘𝑁) ∈ ℕ0s ↔ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
13	3, 12	orbi12d 919	. . . 4 ⊢ (𝑁 ∈ No → ((𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s ))))
14		3orcoma 1093	. . . . 5 ⊢ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 = 0s ∨ 𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs))
15		3orass 1090	. . . . 5 ⊢ ((𝑁 = 0s ∨ 𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)))
16		orcom 871	. . . . . 6 ⊢ ((𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ ((𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ∨ 𝑁 = 0s ))
17		orordir 930	. . . . . 6 ⊢ (((𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs) ∨ 𝑁 = 0s ) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
18	16, 17	bitri 275	. . . . 5 ⊢ ((𝑁 = 0s ∨ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )))
19	14, 15, 18	3bitrri 298	. . . 4 ⊢ (((𝑁 ∈ ℕs ∨ 𝑁 = 0s ) ∨ (( -us ‘𝑁) ∈ ℕs ∨ 𝑁 = 0s )) ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs))
20	13, 19	bitr2di 288	. . 3 ⊢ (𝑁 ∈ No → ((𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs) ↔ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
21	20	pm5.32i 574	. 2 ⊢ ((𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs)) ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
22	1, 21	bitr4i 278	1 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114  ‘cfv 6499   No csur 27603   0_s c0s 27797   -u_s cnegs 28011  ℕ_0scn0s 28304  ℕ_scnns 28305  ℤ_sczs 28370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-n0s 28306  df-nns 28307  df-zs 28371

This theorem is referenced by:  elnnzs  28393  elznns  28394