Theorem zcuts0 28400

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 28390	. 2 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
2		n0on 28328	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
3		elons 28245	. . . . . . . 8 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R ‘𝐴) = ∅))
4	3	simprbi 497	. . . . . . 7 ⊢ (𝐴 ∈ Ons → ( R ‘𝐴) = ∅)
5	2, 4	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ( R ‘𝐴) = ∅)
6	5	a1i 11	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ∈ ℕ0s → ( R ‘𝐴) = ∅))
7		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 ∈ No )
8	7	negscld 28029	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) ∈ No )
9		negleft 28050	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ No → ( L ‘( -us ‘( -us ‘𝐴))) = ( -us “ ( R ‘( -us ‘𝐴))))
10	8, 9	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( L ‘( -us ‘( -us ‘𝐴))) = ( -us “ ( R ‘( -us ‘𝐴))))
11		negnegs 28036	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
12	11	fveq2d 6844	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( L ‘( -us ‘( -us ‘𝐴))) = ( L ‘𝐴))
13	12	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( L ‘( -us ‘( -us ‘𝐴))) = ( L ‘𝐴))
14		n0on 28328	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) ∈ Ons)
15		elons 28245	. . . . . . . . . . . 12 ⊢ (( -us ‘𝐴) ∈ Ons ↔ (( -us ‘𝐴) ∈ No ∧ ( R ‘( -us ‘𝐴)) = ∅))
16	15	simprbi 497	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ Ons → ( R ‘( -us ‘𝐴)) = ∅)
17	14, 16	syl 17	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( R ‘( -us ‘𝐴)) = ∅)
18	17	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( R ‘( -us ‘𝐴)) = ∅)
19	18	imaeq2d 6025	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ ( R ‘( -us ‘𝐴))) = ( -us “ ∅))
20		ima0 6042	. . . . . . . 8 ⊢ ( -us “ ∅) = ∅
21	19, 20	eqtrdi 2787	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ ( R ‘( -us ‘𝐴))) = ∅)
22	10, 13, 21	3eqtr3d 2779	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( L ‘𝐴) = ∅)
23	22	ex 412	. . . . 5 ⊢ (𝐴 ∈ No → (( -us ‘𝐴) ∈ ℕ0s → ( L ‘𝐴) = ∅))
24	6, 23	orim12d 967	. . . 4 ⊢ (𝐴 ∈ No → ((𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s) → (( R ‘𝐴) = ∅ ∨ ( L ‘𝐴) = ∅)))
25	24	imp 406	. . 3 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) → (( R ‘𝐴) = ∅ ∨ ( L ‘𝐴) = ∅))
26	25	orcomd 872	. 2 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) → (( L ‘𝐴) = ∅ ∨ ( R ‘𝐴) = ∅))
27	1, 26	sylbi 217	1 ⊢ (𝐴 ∈ ℤs → (( L ‘𝐴) = ∅ ∨ ( R ‘𝐴) = ∅))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∅c0 4273   “ cima 5634  ‘cfv 6498   No csur 27603   L cleft 27817   R cright 27818   -u_s cnegs 28011  On_scons 28243  ℕ_0scn0s 28304  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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  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 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  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-ons 28244  df-n0s 28306  df-nns 28307  df-zs 28371

This theorem is referenced by: (None)