Theorem 1reno 28690

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 28003	. 2 ⊢ 1s ∈ No
2		2nns 28611	. . . 4 ⊢ 2s ∈ ℕs
3		2no 28612	. . . . . . . . 9 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → 2s ∈ No )
5	4	negscld 28230	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) ∈ No )
6		0no 28002	. . . . . . . 8 ⊢ 0s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s ∈ No )
8	1	a1i 11	. . . . . . 7 ⊢ (⊤ → 1s ∈ No )
9		nnsgt0 28532	. . . . . . . . 9 ⊢ (2s ∈ ℕs → 0s <s 2s)
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ 0s <s 2s
11	4	lt0negs2d 28244	. . . . . . . 8 ⊢ (⊤ → ( 0s <s 2s ↔ ( -us ‘2s) <s 0s ))
12	10, 11	mpbii 236	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) <s 0s )
13		0lt1s 28005	. . . . . . . 8 ⊢ 0s <s 1s
14	13	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s <s 1s )
15	5, 7, 8, 12, 14	ltstrd 27927	. . . . . 6 ⊢ (⊤ → ( -us ‘2s) <s 1s )
16	15	mptru 1577	. . . . 5 ⊢ ( -us ‘2s) <s 1s
17	8	ltsp1d 28208	. . . . . . 7 ⊢ (⊤ → 1s <s ( 1s +s 1s ))
18	17	mptru 1577	. . . . . 6 ⊢ 1s <s ( 1s +s 1s )
19		1p1e2s 28609	. . . . . 6 ⊢ ( 1s +s 1s ) = 2s
20	18, 19	breqtri 5136	. . . . 5 ⊢ 1s <s 2s
21	16, 20	pm3.2i 475	. . . 4 ⊢ (( -us ‘2s) <s 1s ∧ 1s <s 2s)
22		fveq2 6881	. . . . . . 7 ⊢ (𝑛 = 2s → ( -us ‘𝑛) = ( -us ‘2s))
23	22	breq1d 5119	. . . . . 6 ⊢ (𝑛 = 2s → (( -us ‘𝑛) <s 1s ↔ ( -us ‘2s) <s 1s ))
24		breq2 5113	. . . . . 6 ⊢ (𝑛 = 2s → ( 1s <s 𝑛 ↔ 1s <s 2s))
25	23, 24	anbi12d 643	. . . . 5 ⊢ (𝑛 = 2s → ((( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛) ↔ (( -us ‘2s) <s 1s ∧ 1s <s 2s)))
26	25	rspcev 3581	. . . 4 ⊢ ((2s ∈ ℕs ∧ (( -us ‘2s) <s 1s ∧ 1s <s 2s)) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛))
27	2, 21, 26	mp2an 704	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛)
28		1nns 28542	. . . . 5 ⊢ 1s ∈ ℕs
29		lesid 27931	. . . . . 6 ⊢ ( 1s ∈ No → 1s ≤s 1s )
30	1, 29	ax-mp 5	. . . . 5 ⊢ 1s ≤s 1s
31		oveq2 7418	. . . . . . . 8 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = ( 1s /su 1s ))
32		divs1 28397	. . . . . . . . 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 5119	. . . . . 6 ⊢ (𝑛 = 1s → (( 1s /su 𝑛) ≤s 1s ↔ 1s ≤s 1s ))
36	35	rspcev 3581	. . . . 5 ⊢ (( 1s ∈ ℕs ∧ 1s ≤s 1s ) → ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
37	28, 30, 36	mp2an 704	. . . 4 ⊢ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s
38		left1s 28088	. . . . . . . 8 ⊢ ( L ‘ 1s ) = { 0s }
39		right1s 28089	. . . . . . . 8 ⊢ ( R ‘ 1s ) = ∅
40	38, 39	uneq12i 4120	. . . . . . 7 ⊢ (( L ‘ 1s ) ∪ ( R ‘ 1s )) = ({ 0s } ∪ ∅)
41		un0 4351	. . . . . . 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 3477	. . . . . 6 ⊢ 0s ∈ V
45		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑥𝑂 = 0s → ( 1s -s 𝑥𝑂) = ( 1s -s 0s ))
46		subsid1 28261	. . . . . . . . . . . 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 6885	. . . . . . . . 9 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s -s 𝑥𝑂)) = (abss‘ 1s ))
50	7, 8, 14	ltlesd 27937	. . . . . . . . . . 11 ⊢ (⊤ → 0s ≤s 1s )
51	50	mptru 1577	. . . . . . . . . 10 ⊢ 0s ≤s 1s
52		abssid 28434	. . . . . . . . . 10 ⊢ (( 1s ∈ No ∧ 0s ≤s 1s ) → (abss‘ 1s ) = 1s )
53	1, 51, 52	mp2an 704	. . . . . . . . 9 ⊢ (abss‘ 1s ) = 1s
54	49, 53	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s -s 𝑥𝑂)) = 1s )
55	54	breq2d 5121	. . . . . . 7 ⊢ (𝑥𝑂 = 0s → (( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s 1s ))
56	55	rexbidv 3189	. . . . . 6 ⊢ (𝑥𝑂 = 0s → (∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s ))
57	44, 56	ralsn 4647	. . . . 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 475	. 2 ⊢ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛) ∧ ∀𝑥𝑂 ∈ (( L ‘ 1s ) ∪ ( R ‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)))
61		elreno2 28688	. 2 ⊢ ( 1s ∈ ℝs ↔ ( 1s ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘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 723	1 ⊢ 1s ∈ ℝs

Syntax hints:   ∧ wa 400   = wceq 1570  ⊤wtru 1571   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∪ cun 3903  ∅c0 4286  {csn 4589   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410   No csur 27804   <s clts 27805   ≤s cles 27908   0_s c0s 27998   1_s c1s 27999   L cleft 28018   R cright 28019   +_s cadds 28152   -u_s cnegs 28212   -_s csubs 28213   /_su cdivs 28380  abs_scabss 28430  ℕ_scnns 28506  2_sc2s 28603  ℝ_screno 28682

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 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-dc 10425

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 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-oadd 8453  df-nadd 8648  df-no 27807  df-lts 27808  df-bday 27809  df-les 27909  df-slts 27951  df-cuts 27953  df-0s 28000  df-1s 28001  df-made 28020  df-old 28021  df-left 28023  df-right 28024  df-norec 28131  df-norec2 28142  df-adds 28153  df-negs 28214  df-subs 28215  df-muls 28300  df-divs 28381  df-abss 28431  df-n0s 28507  df-nns 28508  df-2s 28604  df-reno 28683

This theorem is referenced by: (None)