Theorem 1reno 28762

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 28075	. 2 ⊢ 1s ∈ No
2		2nns 28683	. . . 4 ⊢ 2s ∈ ℕs
3		2no 28684	. . . . . . . . 9 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → 2s ∈ No )
5	4	negscld 28302	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) ∈ No )
6		0no 28074	. . . . . . . 8 ⊢ 0s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s ∈ No )
8	1	a1i 11	. . . . . . 7 ⊢ (⊤ → 1s ∈ No )
9		nnsgt0 28604	. . . . . . . . 9 ⊢ (2s ∈ ℕs → 0s <s 2s)
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ 0s <s 2s
11	4	lt0negs2d 28316	. . . . . . . 8 ⊢ (⊤ → ( 0s <s 2s ↔ ( -us ‘2s) <s 0s ))
12	10, 11	mpbii 236	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) <s 0s )
13		0lt1s 28077	. . . . . . . 8 ⊢ 0s <s 1s
14	13	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s <s 1s )
15	5, 7, 8, 12, 14	ltstrd 27999	. . . . . 6 ⊢ (⊤ → ( -us ‘2s) <s 1s )
16	15	mptru 1577	. . . . 5 ⊢ ( -us ‘2s) <s 1s
17	8	ltsp1d 28280	. . . . . . 7 ⊢ (⊤ → 1s <s ( 1s +s 1s ))
18	17	mptru 1577	. . . . . 6 ⊢ 1s <s ( 1s +s 1s )
19		1p1e2s 28681	. . . . . 6 ⊢ ( 1s +s 1s ) = 2s
20	18, 19	breqtri 5130	. . . . 5 ⊢ 1s <s 2s
21	16, 20	pm3.2i 476	. . . 4 ⊢ (( -us ‘2s) <s 1s ∧ 1s <s 2s)
22		fveq2 6878	. . . . . . 7 ⊢ (𝑛 = 2s → ( -us ‘𝑛) = ( -us ‘2s))
23	22	breq1d 5113	. . . . . 6 ⊢ (𝑛 = 2s → (( -us ‘𝑛) <s 1s ↔ ( -us ‘2s) <s 1s ))
24		breq2 5107	. . . . . 6 ⊢ (𝑛 = 2s → ( 1s <s 𝑛 ↔ 1s <s 2s))
25	23, 24	anbi12d 644	. . . . 5 ⊢ (𝑛 = 2s → ((( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛) ↔ (( -us ‘2s) <s 1s ∧ 1s <s 2s)))
26	25	rspcev 3576	. . . 4 ⊢ ((2s ∈ ℕs ∧ (( -us ‘2s) <s 1s ∧ 1s <s 2s)) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛))
27	2, 21, 26	mp2an 705	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛)
28		1nns 28614	. . . . 5 ⊢ 1s ∈ ℕs
29		lesid 28003	. . . . . 6 ⊢ ( 1s ∈ No → 1s ≤s 1s )
30	1, 29	ax-mp 5	. . . . 5 ⊢ 1s ≤s 1s
31		oveq2 7421	. . . . . . . 8 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = ( 1s /su 1s ))
32		divs1 28469	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
33	1, 32	ax-mp 5	. . . . . . . 8 ⊢ ( 1s /su 1s ) = 1s
34	31, 33	eqtrdi 2811	. . . . . . 7 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = 1s )
35	34	breq1d 5113	. . . . . 6 ⊢ (𝑛 = 1s → (( 1s /su 𝑛) ≤s 1s ↔ 1s ≤s 1s ))
36	35	rspcev 3576	. . . . 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 28160	. . . . . . . 8 ⊢ ( L ‘ 1s ) = { 0s }
39		right1s 28161	. . . . . . . 8 ⊢ ( R ‘ 1s ) = ∅
40	38, 39	uneq12i 4113	. . . . . . 7 ⊢ (( L ‘ 1s ) ∪ ( R ‘ 1s )) = ({ 0s } ∪ ∅)
41		un0 4344	. . . . . . 7 ⊢ ({ 0s } ∪ ∅) = { 0s }
42	40, 41	eqtri 2783	. . . . . 6 ⊢ (( L ‘ 1s ) ∪ ( R ‘ 1s )) = { 0s }
43	42	raleqi 3317	. . . . 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 3472	. . . . . 6 ⊢ 0s ∈ V
45		oveq2 7421	. . . . . . . . . . 11 ⊢ (𝑥𝑂 = 0s → ( 1s -s 𝑥𝑂) = ( 1s -s 0s ))
46		subsid1 28333	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s -s 0s ) = 1s )
47	1, 46	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s -s 0s ) = 1s
48	45, 47	eqtrdi 2811	. . . . . . . . . 10 ⊢ (𝑥𝑂 = 0s → ( 1s -s 𝑥𝑂) = 1s )
49	48	fveq2d 6882	. . . . . . . . 9 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s -s 𝑥𝑂)) = (abss‘ 1s ))
50	7, 8, 14	ltlesd 28009	. . . . . . . . . . 11 ⊢ (⊤ → 0s ≤s 1s )
51	50	mptru 1577	. . . . . . . . . 10 ⊢ 0s ≤s 1s
52		abssid 28506	. . . . . . . . . 10 ⊢ (( 1s ∈ No ∧ 0s ≤s 1s ) → (abss‘ 1s ) = 1s )
53	1, 51, 52	mp2an 705	. . . . . . . . 9 ⊢ (abss‘ 1s ) = 1s
54	49, 53	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s -s 𝑥𝑂)) = 1s )
55	54	breq2d 5115	. . . . . . 7 ⊢ (𝑥𝑂 = 0s → (( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s 1s ))
56	55	rexbidv 3186	. . . . . 6 ⊢ (𝑥𝑂 = 0s → (∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s ))
57	44, 56	ralsn 4642	. . . . 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 (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛) ∧ ∀𝑥𝑂 ∈ (( L ‘ 1s ) ∪ ( R ‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)))
61		elreno2 28760	. 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 724	1 ⊢ 1s ∈ ℝs

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∪ cun 3897  ∅c0 4279  {csn 4584   class class class wbr 5103  ‘cfv 6533  (class class class)co 7413   No csur 27876   <s clts 27877   ≤s cles 27980   0_s c0s 28070   1_s c1s 28071   L cleft 28090   R cright 28091   +_s cadds 28224   -u_s cnegs 28284   -_s csubs 28285   /_su cdivs 28452  abs_scabss 28502  ℕ_scnns 28578  2_sc2s 28675  ℝ_screno 28754

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-dc 10448

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-nadd 8654  df-no 27879  df-lts 27880  df-bday 27881  df-les 27981  df-slts 28023  df-cuts 28025  df-0s 28072  df-1s 28073  df-made 28092  df-old 28093  df-left 28095  df-right 28096  df-norec 28203  df-norec2 28214  df-adds 28225  df-negs 28286  df-subs 28287  df-muls 28372  df-divs 28453  df-abss 28503  df-n0s 28579  df-nns 28580  df-2s 28676  df-reno 28755

This theorem is used by: (None)