Theorem elzs2 28407

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 28406	. 2 ⊢ (𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕ0s ∨ ( -us ‘𝑁) ∈ ℕ0s)))
2		eln0s 28369	. . . . . 6 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ No → (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s )))
4		eln0s 28369	. . . . . 6 ⊢ (( -us ‘𝑁) ∈ ℕ0s ↔ (( -us ‘𝑁) ∈ ℕs ∨ ( -us ‘𝑁) = 0s ))
5		neg0s 28034	. . . . . . . . 9 ⊢ ( -us ‘ 0s ) = 0s
6	5	eqeq2i 2750	. . . . . . . 8 ⊢ (( -us ‘𝑁) = ( -us ‘ 0s ) ↔ ( -us ‘𝑁) = 0s )
7		0no 27817	. . . . . . . . 9 ⊢ 0s ∈ No
8		negs11 28057	. . . . . . . . 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 6500   No csur 27619   0_s c0s 27813   -u_s cnegs 28027  ℕ_0scn0s 28320  ℕ_scnns 28321  ℤ_sczs 28386

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-n0s 28322  df-nns 28323  df-zs 28387

This theorem is referenced by:  elnnzs  28409  elznns  28410