Theorem 1reno 28876

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 28189	. 2 ⊢ 1s ∈ No
2		2nns 28797	. . . 4 ⊢ 2s ∈ ℕs
3		2no 28798	. . . . . . . . 9 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → 2s ∈ No )
5	4	negscld 28416	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) ∈ No )
6		0no 28188	. . . . . . . 8 ⊢ 0s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s ∈ No )
8	1	a1i 11	. . . . . . 7 ⊢ (⊤ → 1s ∈ No )
9		nnsgt0 28718	. . . . . . . . 9 ⊢ (2s ∈ ℕs → 0s <s 2s)
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ 0s <s 2s
11	4	lt0negs2d 28430	. . . . . . . 8 ⊢ (⊤ → ( 0s <s 2s ↔ ( -us ‘2s) <s 0s ))
12	10, 11	mpbii 236	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) <s 0s )
13		0lt1s 28191	. . . . . . . 8 ⊢ 0s <s 1s
14	13	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s <s 1s )
15	5, 7, 8, 12, 14	ltstrd 28113	. . . . . 6 ⊢ (⊤ → ( -us ‘2s) <s 1s )
16	15	mptru 1577	. . . . 5 ⊢ ( -us ‘2s) <s 1s
17	8	ltsp1d 28394	. . . . . . 7 ⊢ (⊤ → 1s <s ( 1s +s 1s ))
18	17	mptru 1577	. . . . . 6 ⊢ 1s <s ( 1s +s 1s )
19		1p1e2s 28795	. . . . . 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 6883	. . . . . . 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 3577	. . . 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 28728	. . . . 5 ⊢ 1s ∈ ℕs
29		lesid 28117	. . . . . 6 ⊢ ( 1s ∈ No → 1s ≤s 1s )
30	1, 29	ax-mp 5	. . . . 5 ⊢ 1s ≤s 1s
31		oveq2 7426	. . . . . . . 8 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = ( 1s /su 1s ))
32		divs1 28583	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
33	1, 32	ax-mp 5	. . . . . . . 8 ⊢ ( 1s /su 1s ) = 1s
34	31, 33	eqtrdi 2812	. . . . . . 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 3577	. . . . 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 28274	. . . . . . . 8 ⊢ ( L ‘ 1s ) = { 0s }
39		right1s 28275	. . . . . . . 8 ⊢ ( R ‘ 1s ) = ∅
40	38, 39	uneq12i 4113	. . . . . . 7 ⊢ (( L ‘ 1s ) ∪ ( R ‘ 1s )) = ({ 0s } ∪ ∅)
41		un0 4344	. . . . . . 7 ⊢ ({ 0s } ∪ ∅) = { 0s }
42	40, 41	eqtri 2784	. . . . . 6 ⊢ (( L ‘ 1s ) ∪ ( R ‘ 1s )) = { 0s }
43	42	raleqi 3318	. . . . 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 3473	. . . . . 6 ⊢ 0s ∈ V
45		oveq2 7426	. . . . . . . . . . 11 ⊢ (𝑥𝑂 = 0s → ( 1s -s 𝑥𝑂) = ( 1s -s 0s ))
46		subsid1 28447	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s -s 0s ) = 1s )
47	1, 46	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s -s 0s ) = 1s
48	45, 47	eqtrdi 2812	. . . . . . . . . 10 ⊢ (𝑥𝑂 = 0s → ( 1s -s 𝑥𝑂) = 1s )
49	48	fveq2d 6887	. . . . . . . . 9 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s -s 𝑥𝑂)) = (abss‘ 1s ))
50	7, 8, 14	ltlesd 28123	. . . . . . . . . . 11 ⊢ (⊤ → 0s ≤s 1s )
51	50	mptru 1577	. . . . . . . . . 10 ⊢ 0s ≤s 1s
52		abssid 28620	. . . . . . . . . 10 ⊢ (( 1s ∈ No ∧ 0s ≤s 1s ) → (abss‘ 1s ) = 1s )
53	1, 51, 52	mp2an 705	. . . . . . . . 9 ⊢ (abss‘ 1s ) = 1s
54	49, 53	eqtrdi 2812	. . . . . . . 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 3187	. . . . . 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 28874	. 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 3077  ∃wrex 3087   ∪ cun 3897  ∅c0 4279  {csn 4584   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418   No csur 27990   <s clts 27991   ≤s cles 28094   0_s c0s 28184   1_s c1s 28185   L cleft 28204   R cright 28205   +_s cadds 28338   -u_s cnegs 28398   -_s csubs 28399   /_su cdivs 28566  abs_scabss 28616  ℕ_scnns 28692  2_sc2s 28789  ℝ_screno 28868

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-dc 10517

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-nadd 8668  df-no 27993  df-lts 27994  df-bday 27995  df-les 28095  df-slts 28137  df-cuts 28139  df-0s 28186  df-1s 28187  df-made 28206  df-old 28207  df-left 28209  df-right 28210  df-norec 28317  df-norec2 28328  df-adds 28339  df-negs 28400  df-subs 28401  df-muls 28486  df-divs 28567  df-abss 28617  df-n0s 28693  df-nns 28694  df-2s 28790  df-reno 28869

This theorem is used by: (None)