Theorem zcuts0 28579

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 28569	. 2 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
2		n0on 28507	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
3		elons 28424	. . . . . . . 8 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R ‘𝐴) = ∅))
4	3	simprbi 502	. . . . . . 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 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 ∈ No )
8	7	negscld 28208	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) ∈ No )
9		negleft 28229	. . . . . . . 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 28215	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
12	11	fveq2d 6887	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( L ‘( -us ‘( -us ‘𝐴))) = ( L ‘𝐴))
13	12	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( L ‘( -us ‘( -us ‘𝐴))) = ( L ‘𝐴))
14		n0on 28507	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) ∈ Ons)
15		elons 28424	. . . . . . . . . . . 12 ⊢ (( -us ‘𝐴) ∈ Ons ↔ (( -us ‘𝐴) ∈ No ∧ ( R ‘( -us ‘𝐴)) = ∅))
16	15	simprbi 502	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ Ons → ( R ‘( -us ‘𝐴)) = ∅)
17	14, 16	syl 18	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( R ‘( -us ‘𝐴)) = ∅)
18	17	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( R ‘( -us ‘𝐴)) = ∅)
19	18	imaeq2d 6064	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ ( R ‘( -us ‘𝐴))) = ( -us “ ∅))
20		ima0 6081	. . . . . . . 8 ⊢ ( -us “ ∅) = ∅
21	19, 20	eqtrdi 2814	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ ( R ‘( -us ‘𝐴))) = ∅)
22	10, 13, 21	3eqtr3d 2806	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( L ‘𝐴) = ∅)
23	22	ex 417	. . . . 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 411	. . 3 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) → (( R ‘𝐴) = ∅ ∨ ( L ‘𝐴) = ∅))
26	25	orcomd 884	. 2 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) → (( L ‘𝐴) = ∅ ∨ ( R ‘𝐴) = ∅))
27	1, 26	sylbi 220	1 ⊢ (𝐴 ∈ ℤs → (( L ‘𝐴) = ∅ ∨ ( R ‘𝐴) = ∅))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∅c0 4287   “ cima 5666  ‘cfv 6538   No csur 27782   L cleft 27996   R cright 27997   -u_s cnegs 28190  On_scons 28422  ℕ_0scn0s 28483  ℤ_sczs 28549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27785  df-lts 27786  df-bday 27787  df-les 27887  df-slts 27929  df-cuts 27931  df-0s 27978  df-1s 27979  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec 28109  df-norec2 28120  df-adds 28131  df-negs 28192  df-subs 28193  df-ons 28423  df-n0s 28485  df-nns 28486  df-zs 28550

This theorem is referenced by: (None)