Theorem zcuts0 28404

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 28394	. 2 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
2		n0on 28332	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
3		elons 28249	. . . . . . . 8 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R ‘𝐴) = ∅))
4	3	simprbi 496	. . . . . . 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 28033	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) ∈ No )
9		negleft 28054	. . . . . . . 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 28040	. . . . . . . . 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 28332	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) ∈ Ons)
15		elons 28249	. . . . . . . . . . . 12 ⊢ (( -us ‘𝐴) ∈ Ons ↔ (( -us ‘𝐴) ∈ No ∧ ( R ‘( -us ‘𝐴)) = ∅))
16	15	simprbi 496	. . . . . . . . . . 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 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 966	. . . 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 871	. 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 847   = wceq 1541   ∈ wcel 2113  ∅c0 4285   “ cima 5627  ‘cfv 6492   No csur 27607   L cleft 27821   R cright 27822   -u_s cnegs 28015  On_scons 28247  ℕ_0scn0s 28308  ℤ_sczs 28374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-nadd 8594  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27934  df-norec2 27945  df-adds 27956  df-negs 28017  df-subs 28018  df-ons 28248  df-n0s 28310  df-nns 28311  df-zs 28375

This theorem is referenced by: (None)