Theorem 1reno 28940

Description: Surreal one is a surreal real. (Contributed by Scott Fenton, 18-Feb-2026.)

Assertion

Ref	Expression
1reno	⊢ 1s ∈ ℝs

Proof of Theorem 1reno

Dummy variables 𝑛 𝑥_𝑂 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1no 28253	. 2 ⊢ 1s ∈ No
2		2nns 28861	. . . 4 ⊢ 2s ∈ ℕs
3		2no 28862	. . . . . . . . 9 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → 2s ∈ No)
5	4	negscld 28480	. . . . . . 7 ⊢ (⊤ → ( -s‘2s) ∈ No)
6		0no 28252	. . . . . . . 8 ⊢ 0s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s ∈ No)
8	1	a1i 11	. . . . . . 7 ⊢ (⊤ → 1s ∈ No)
9		nnsgt0 28782	. . . . . . . . 9 ⊢ (2s ∈ ℕs → 0s <s 2s)
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ 0s <s 2s
11	4	lt0negs2d 28494	. . . . . . . 8 ⊢ (⊤ → ( 0s <s 2s ↔ ( -s‘2s) <s 0s ))
12	10, 11	mpbii 236	. . . . . . 7 ⊢ (⊤ → ( -s‘2s) <s 0s )
13		0lt1s 28255	. . . . . . . 8 ⊢ 0s <s 1s
14	13	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s <s 1s )
15	5, 7, 8, 12, 14	ltstrd 28177	. . . . . 6 ⊢ (⊤ → ( -s‘2s) <s 1s )
16	15	mptru 1577	. . . . 5 ⊢ ( -s‘2s) <s 1s
17	8	ltsp1d 28458	. . . . . . 7 ⊢ (⊤ → 1s <s ( 1s +s 1s ))
18	17	mptru 1577	. . . . . 6 ⊢ 1s <s ( 1s +s 1s )
19		1p1e2s 28859	. . . . . 6 ⊢ ( 1s +s 1s ) = 2s
20	18, 19	breqtri 5136	. . . . 5 ⊢ 1s <s 2s
21	16, 20	pm3.2i 476	. . . 4 ⊢ (( -s‘2s) <s 1s ∧ 1s <s 2s)
22		fveq2 6916	. . . . . . 7 ⊢ (𝑛 = 2s → ( -s‘𝑛) = ( -s‘2s))
23	22	breq1d 5118	. . . . . 6 ⊢ (𝑛 = 2s → (( -s‘𝑛) <s 1s ↔ ( -s‘2s) <s 1s ))
24		breq2 5112	. . . . . 6 ⊢ (𝑛 = 2s → ( 1s <s 𝑛 ↔ 1s <s 2s))
25	23, 24	anbi12d 644	. . . . 5 ⊢ (𝑛 = 2s → ((( -s‘𝑛) <s 1s ∧ 1s <s 𝑛) ↔ (( -s‘2s) <s 1s ∧ 1s <s 2s)))
26	25	rspcev 3580	. . . 4 ⊢ ((2s ∈ ℕs ∧ (( -s‘2s) <s 1s ∧ 1s <s 2s)) → ∃𝑛 ∈ ℕs (( -s‘𝑛) <s 1s ∧ 1s <s 𝑛))
27	2, 21, 26	mp2an 705	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -s‘𝑛) <s 1s ∧ 1s <s 𝑛)
28		1nns 28792	. . . . 5 ⊢ 1s ∈ ℕs
29		lesid 28181	. . . . . 6 ⊢ ( 1s ∈ No → 1s ≤s 1s )
30	1, 29	ax-mp 5	. . . . 5 ⊢ 1s ≤s 1s
31		oveq2 7459	. . . . . . . 8 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = ( 1s /su 1s ))
32		divs1 28647	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
33	1, 32	ax-mp 5	. . . . . . . 8 ⊢ ( 1s /su 1s ) = 1s
34	31, 33	eqtrdi 2814	. . . . . . 7 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = 1s )
35	34	breq1d 5118	. . . . . 6 ⊢ (𝑛 = 1s → (( 1s /su 𝑛) ≤s 1s ↔ 1s ≤s 1s ))
36	35	rspcev 3580	. . . . 5 ⊢ (( 1s ∈ ℕs ∧ 1s ≤s 1s ) → ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
37	28, 30, 36	mp2an 705	. . . 4 ⊢ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s
38		left1s 28338	. . . . . . . 8 ⊢ (L‘ 1s ) = { 0s }
39		right1s 28339	. . . . . . . 8 ⊢ (R‘ 1s ) = ∅
40	38, 39	uneq12i 4168	. . . . . . 7 ⊢ ((L‘ 1s ) ∪ (R‘ 1s )) = ({ 0s } ∪ ∅)
41		un0 4347	. . . . . . 7 ⊢ ({ 0s } ∪ ∅) = { 0s }
42	40, 41	eqtri 2786	. . . . . 6 ⊢ ((L‘ 1s ) ∪ (R‘ 1s )) = { 0s }
43	42	raleqi 3321	. . . . 5 ⊢ (∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ∀𝑥𝑂 ∈ { 0s }∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)))
44	6	elexi 3476	. . . . . 6 ⊢ 0s ∈ V
45		oveq2 7459	. . . . . . . . . . 11 ⊢ (𝑥𝑂 = 0s → ( 1s −s 𝑥𝑂) = ( 1s −s 0s ))
46		subsid1 28511	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s −s 0s ) = 1s )
47	1, 46	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s −s 0s ) = 1s
48	45, 47	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝑥𝑂 = 0s → ( 1s −s 𝑥𝑂) = 1s )
49	48	fveq2d 6920	. . . . . . . . 9 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s −s 𝑥𝑂)) = (abss‘ 1s ))
50	7, 8, 14	ltlesd 28187	. . . . . . . . . . 11 ⊢ (⊤ → 0s ≤s 1s )
51	50	mptru 1577	. . . . . . . . . 10 ⊢ 0s ≤s 1s
52		abssid 28684	. . . . . . . . . 10 ⊢ (( 1s ∈ No ∧ 0s ≤s 1s ) → (abss‘ 1s ) = 1s )
53	1, 51, 52	mp2an 705	. . . . . . . . 9 ⊢ (abss‘ 1s ) = 1s
54	49, 53	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s −s 𝑥𝑂)) = 1s )
55	54	breq2d 5120	. . . . . . 7 ⊢ (𝑥𝑂 = 0s → (( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s 1s ))
56	55	rexbidv 3190	. . . . . 6 ⊢ (𝑥𝑂 = 0s → (∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s ))
57	44, 56	ralsn 4643	. . . . 5 ⊢ (∀𝑥𝑂 ∈ { 0s }∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
58	43, 57	bitri 278	. . . 4 ⊢ (∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
59	37, 58	mpbir 234	. . 3 ⊢ ∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂))
60	27, 59	pm3.2i 476	. 2 ⊢ (∃𝑛 ∈ ℕs (( -s‘𝑛) <s 1s ∧ 1s <s 𝑛) ∧ ∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)))
61		elreno2 28938	. 2 ⊢ ( 1s ∈ ℝs ↔ ( 1s ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘𝑛) <s 1s ∧ 1s <s 𝑛) ∧ ∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)))))
62	1, 60, 61	mpbir2an 724	1 ⊢ 1s ∈ ℝs

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ∪ cun 4152  ∅c0 4281  {csn 4588   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Nocsur 28054   <s clts 28055   ≤s cles 28158   0_s c0s 28248   1_s c1s 28249  Lcleft 28268  Rcright 28269   +_s cadds 28402   -_scnegs 28462   −_s csubs 28463   /_su cdivs 28630  abs_scabss 28680  ℕ_scnns 28756  2_sc2s 28853  ℝ_screno 28932

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-dc 10563

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-oadd 8518  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-1s 28251  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-muls 28550  df-divs 28631  df-abss 28681  df-n0s 28757  df-nns 28758  df-2s 28854  df-reno 28933

This theorem is used by: (None)