Theorem zcuts0 28414

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 28404	. 2 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
2		n0on 28342	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
3		elons 28259	. . . . . . . 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 28043	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) ∈ No )
9		negleft 28064	. . . . . . . 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 28050	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ( -us ‘( -us ‘𝐴)) = 𝐴)
12	11	fveq2d 6838	. . . . . . . 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 28342	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) ∈ Ons)
15		elons 28259	. . . . . . . . . . . 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 6019	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ ( R ‘( -us ‘𝐴))) = ( -us “ ∅))
20		ima0 6036	. . . . . . . 8 ⊢ ( -us “ ∅) = ∅
21	19, 20	eqtrdi 2788	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ ( R ‘( -us ‘𝐴))) = ∅)
22	10, 13, 21	3eqtr3d 2780	. . . . . 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 4274   “ cima 5627  ‘cfv 6492   No csur 27617   L cleft 27831   R cright 27832   -u_s cnegs 28025  On_scons 28257  ℕ_0scn0s 28318  ℤ_sczs 28384

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-1s 27814  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec 27944  df-norec2 27955  df-adds 27966  df-negs 28027  df-subs 28028  df-ons 28258  df-n0s 28320  df-nns 28321  df-zs 28385

This theorem is referenced by: (None)