Theorem 1reno 28507

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 27820	. 2 ⊢ 1s ∈ No
2		2nns 28428	. . . 4 ⊢ 2s ∈ ℕs
3		2no 28429	. . . . . . . . 9 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → 2s ∈ No )
5	4	negscld 28047	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) ∈ No )
6		0no 27819	. . . . . . . 8 ⊢ 0s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s ∈ No )
8	1	a1i 11	. . . . . . 7 ⊢ (⊤ → 1s ∈ No )
9		nnsgt0 28349	. . . . . . . . 9 ⊢ (2s ∈ ℕs → 0s <s 2s)
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ 0s <s 2s
11	4	lt0negs2d 28061	. . . . . . . 8 ⊢ (⊤ → ( 0s <s 2s ↔ ( -us ‘2s) <s 0s ))
12	10, 11	mpbii 234	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) <s 0s )
13		0lt1s 27822	. . . . . . . 8 ⊢ 0s <s 1s
14	13	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s <s 1s )
15	5, 7, 8, 12, 14	ltstrd 27745	. . . . . 6 ⊢ (⊤ → ( -us ‘2s) <s 1s )
16	15	mptru 1554	. . . . 5 ⊢ ( -us ‘2s) <s 1s
17	8	ltsp1d 28025	. . . . . . 7 ⊢ (⊤ → 1s <s ( 1s +s 1s ))
18	17	mptru 1554	. . . . . 6 ⊢ 1s <s ( 1s +s 1s )
19		1p1e2s 28426	. . . . . 6 ⊢ ( 1s +s 1s ) = 2s
20	18, 19	breqtri 5097	. . . . 5 ⊢ 1s <s 2s
21	16, 20	pm3.2i 471	. . . 4 ⊢ (( -us ‘2s) <s 1s ∧ 1s <s 2s)
22		fveq2 6827	. . . . . . 7 ⊢ (𝑛 = 2s → ( -us ‘𝑛) = ( -us ‘2s))
23	22	breq1d 5082	. . . . . 6 ⊢ (𝑛 = 2s → (( -us ‘𝑛) <s 1s ↔ ( -us ‘2s) <s 1s ))
24		breq2 5076	. . . . . 6 ⊢ (𝑛 = 2s → ( 1s <s 𝑛 ↔ 1s <s 2s))
25	23, 24	anbi12d 638	. . . . 5 ⊢ (𝑛 = 2s → ((( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛) ↔ (( -us ‘2s) <s 1s ∧ 1s <s 2s)))
26	25	rspcev 3560	. . . 4 ⊢ ((2s ∈ ℕs ∧ (( -us ‘2s) <s 1s ∧ 1s <s 2s)) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛))
27	2, 21, 26	mp2an 698	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛)
28		1nns 28359	. . . . 5 ⊢ 1s ∈ ℕs
29		lesid 27749	. . . . . 6 ⊢ ( 1s ∈ No → 1s ≤s 1s )
30	1, 29	ax-mp 5	. . . . 5 ⊢ 1s ≤s 1s
31		oveq2 7364	. . . . . . . 8 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = ( 1s /su 1s ))
32		divs1 28214	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
33	1, 32	ax-mp 5	. . . . . . . 8 ⊢ ( 1s /su 1s ) = 1s
34	31, 33	eqtrdi 2790	. . . . . . 7 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = 1s )
35	34	breq1d 5082	. . . . . 6 ⊢ (𝑛 = 1s → (( 1s /su 𝑛) ≤s 1s ↔ 1s ≤s 1s ))
36	35	rspcev 3560	. . . . 5 ⊢ (( 1s ∈ ℕs ∧ 1s ≤s 1s ) → ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
37	28, 30, 36	mp2an 698	. . . 4 ⊢ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s
38		left1s 27905	. . . . . . . 8 ⊢ ( L ‘ 1s ) = { 0s }
39		right1s 27906	. . . . . . . 8 ⊢ ( R ‘ 1s ) = ∅
40	38, 39	uneq12i 4096	. . . . . . 7 ⊢ (( L ‘ 1s ) ∪ ( R ‘ 1s )) = ({ 0s } ∪ ∅)
41		un0 4322	. . . . . . 7 ⊢ ({ 0s } ∪ ∅) = { 0s }
42	40, 41	eqtri 2762	. . . . . 6 ⊢ (( L ‘ 1s ) ∪ ( R ‘ 1s )) = { 0s }
43	42	raleqi 3295	. . . . 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 3453	. . . . . 6 ⊢ 0s ∈ V
45		oveq2 7364	. . . . . . . . . . 11 ⊢ (𝑥𝑂 = 0s → ( 1s -s 𝑥𝑂) = ( 1s -s 0s ))
46		subsid1 28078	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s -s 0s ) = 1s )
47	1, 46	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s -s 0s ) = 1s
48	45, 47	eqtrdi 2790	. . . . . . . . . 10 ⊢ (𝑥𝑂 = 0s → ( 1s -s 𝑥𝑂) = 1s )
49	48	fveq2d 6831	. . . . . . . . 9 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s -s 𝑥𝑂)) = (abss‘ 1s ))
50	7, 8, 14	ltlesd 27755	. . . . . . . . . . 11 ⊢ (⊤ → 0s ≤s 1s )
51	50	mptru 1554	. . . . . . . . . 10 ⊢ 0s ≤s 1s
52		abssid 28251	. . . . . . . . . 10 ⊢ (( 1s ∈ No ∧ 0s ≤s 1s ) → (abss‘ 1s ) = 1s )
53	1, 51, 52	mp2an 698	. . . . . . . . 9 ⊢ (abss‘ 1s ) = 1s
54	49, 53	eqtrdi 2790	. . . . . . . 8 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s -s 𝑥𝑂)) = 1s )
55	54	breq2d 5084	. . . . . . 7 ⊢ (𝑥𝑂 = 0s → (( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s 1s ))
56	55	rexbidv 3163	. . . . . 6 ⊢ (𝑥𝑂 = 0s → (∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s ))
57	44, 56	ralsn 4613	. . . . 5 ⊢ (∀𝑥𝑂 ∈ { 0s }∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
58	43, 57	bitri 276	. . . 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 232	. . 3 ⊢ ∀𝑥𝑂 ∈ (( L ‘ 1s ) ∪ ( R ‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂))
60	27, 59	pm3.2i 471	. 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 28505	. 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 717	1 ⊢ 1s ∈ ℝs

Syntax hints:   ∧ wa 396   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ∪ cun 3881  ∅c0 4261  {csn 4555   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356   No csur 27621   <s clts 27622   ≤s cles 27726   0_s c0s 27815   1_s c1s 27816   L cleft 27835   R cright 27836   +_s cadds 27969   -u_s cnegs 28029   -_s csubs 28030   /_su cdivs 28197  abs_scabss 28247  ℕ_scnns 28323  2_sc2s 28420  ℝ_screno 28499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553  ax-dc 10359

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-ot 4564  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-nadd 8592  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031  df-subs 28032  df-muls 28117  df-divs 28198  df-abss 28248  df-n0s 28324  df-nns 28325  df-2s 28421  df-reno 28500

This theorem is referenced by: (None)