Theorem zcuts0 28638

Description: Either the left or right set of a surreal integer is empty. (Contributed by Scott Fenton, 21-Feb-2026.)

Assertion

Ref	Expression
zcuts0	⊢ (𝐴 ∈ ℤs → (( L ‘𝐴) = ∅ ∨ ( R ‘𝐴) = ∅))

Proof of Theorem zcuts0

Step	Hyp	Ref	Expression
1		elzn0s 28628	. 2 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
2		n0on 28566	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
3		elons 28483	. . . . . . . 8 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R ‘𝐴) = ∅))
4	3	simprbi 503	. . . . . . 7 ⊢ (𝐴 ∈ Ons → ( R ‘𝐴) = ∅)
5	2, 4	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ( R ‘𝐴) = ∅)
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ∈ ℕ0s → ( R ‘𝐴) = ∅))
7		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 ∈ No )
8	7	negscld 28267	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) ∈ No )
9		negleft 28288	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ No → ( L ‘( -us ‘( -us ‘𝐴))) = ( -us “ ( R ‘( -us ‘𝐴))))
10	8, 9	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( L ‘( -us ‘( -us ‘𝐴))) = ( -us “ ( R ‘( -us ‘𝐴))))
11		negnegs 28274	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
12	11	fveq2d 6892	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( L ‘( -us ‘( -us ‘𝐴))) = ( L ‘𝐴))
13	12	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( L ‘( -us ‘( -us ‘𝐴))) = ( L ‘𝐴))
14		n0on 28566	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) ∈ Ons)
15		elons 28483	. . . . . . . . . . . 12 ⊢ (( -us ‘𝐴) ∈ Ons ↔ (( -us ‘𝐴) ∈ No ∧ ( R ‘( -us ‘𝐴)) = ∅))
16	15	simprbi 503	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ Ons → ( R ‘( -us ‘𝐴)) = ∅)
17	14, 16	syl 18	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( R ‘( -us ‘𝐴)) = ∅)
18	17	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( R ‘( -us ‘𝐴)) = ∅)
19	18	imaeq2d 6067	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ ( R ‘( -us ‘𝐴))) = ( -us “ ∅))
20		ima0 6084	. . . . . . . 8 ⊢ ( -us “ ∅) = ∅
21	19, 20	eqtrdi 2817	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ ( R ‘( -us ‘𝐴))) = ∅)
22	10, 13, 21	3eqtr3d 2809	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( L ‘𝐴) = ∅)
23	22	ex 418	. . . . 5 ⊢ (𝐴 ∈ No → (( -us ‘𝐴) ∈ ℕ0s → ( L ‘𝐴) = ∅))
24	6, 23	orim12d 979	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s) → (( R ‘𝐴) = ∅ ∨ ( L ‘𝐴) = ∅)))
25	24	imp 412	. . 3 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) → (( R ‘𝐴) = ∅ ∨ ( L ‘𝐴) = ∅))
26	25	orcomd 885	. 2 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) → (( L ‘𝐴) = ∅ ∨ ( R ‘𝐴) = ∅))
27	1, 26	sylbi 220	1 ⊢ (𝐴 ∈ ℤs → (( L ‘𝐴) = ∅ ∨ ( R ‘𝐴) = ∅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∅c0 4289   “ cima 5669  ‘cfv 6543   No csur 27841   L cleft 28055   R cright 28056   -u_s cnegs 28249  On_scons 28481  ℕ_0scn0s 28542  ℤ_sczs 28608

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-nadd 8661  df-no 27844  df-lts 27845  df-bday 27846  df-les 27946  df-slts 27988  df-cuts 27990  df-0s 28037  df-1s 28038  df-made 28057  df-old 28058  df-left 28060  df-right 28061  df-norec 28168  df-norec2 28179  df-adds 28190  df-negs 28251  df-subs 28252  df-ons 28482  df-n0s 28544  df-nns 28545  df-zs 28609

This theorem is used by: (None)