Theorem 1reno 28506

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 27819	. 2 ⊢ 1s ∈ No
2		2nns 28427	. . . 4 ⊢ 2s ∈ ℕs
3		2no 28428	. . . . . . . . 9 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → 2s ∈ No )
5	4	negscld 28046	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) ∈ No )
6		0no 27818	. . . . . . . 8 ⊢ 0s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s ∈ No )
8	1	a1i 11	. . . . . . 7 ⊢ (⊤ → 1s ∈ No )
9		nnsgt0 28348	. . . . . . . . 9 ⊢ (2s ∈ ℕs → 0s <s 2s)
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ 0s <s 2s
11	4	lt0negs2d 28060	. . . . . . . 8 ⊢ (⊤ → ( 0s <s 2s ↔ ( -us ‘2s) <s 0s ))
12	10, 11	mpbii 233	. . . . . . 7 ⊢ (⊤ → ( -us ‘2s) <s 0s )
13		0lt1s 27821	. . . . . . . 8 ⊢ 0s <s 1s
14	13	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s <s 1s )
15	5, 7, 8, 12, 14	ltstrd 27744	. . . . . 6 ⊢ (⊤ → ( -us ‘2s) <s 1s )
16	15	mptru 1549	. . . . 5 ⊢ ( -us ‘2s) <s 1s
17	8	ltsp1d 28024	. . . . . . 7 ⊢ (⊤ → 1s <s ( 1s +s 1s ))
18	17	mptru 1549	. . . . . 6 ⊢ 1s <s ( 1s +s 1s )
19		1p1e2s 28425	. . . . . 6 ⊢ ( 1s +s 1s ) = 2s
20	18, 19	breqtri 5111	. . . . 5 ⊢ 1s <s 2s
21	16, 20	pm3.2i 470	. . . 4 ⊢ (( -us ‘2s) <s 1s ∧ 1s <s 2s)
22		fveq2 6835	. . . . . . 7 ⊢ (𝑛 = 2s → ( -us ‘𝑛) = ( -us ‘2s))
23	22	breq1d 5096	. . . . . 6 ⊢ (𝑛 = 2s → (( -us ‘𝑛) <s 1s ↔ ( -us ‘2s) <s 1s ))
24		breq2 5090	. . . . . 6 ⊢ (𝑛 = 2s → ( 1s <s 𝑛 ↔ 1s <s 2s))
25	23, 24	anbi12d 633	. . . . 5 ⊢ (𝑛 = 2s → ((( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛) ↔ (( -us ‘2s) <s 1s ∧ 1s <s 2s)))
26	25	rspcev 3565	. . . 4 ⊢ ((2s ∈ ℕs ∧ (( -us ‘2s) <s 1s ∧ 1s <s 2s)) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛))
27	2, 21, 26	mp2an 693	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 1s ∧ 1s <s 𝑛)
28		1nns 28358	. . . . 5 ⊢ 1s ∈ ℕs
29		lesid 27748	. . . . . 6 ⊢ ( 1s ∈ No → 1s ≤s 1s )
30	1, 29	ax-mp 5	. . . . 5 ⊢ 1s ≤s 1s
31		oveq2 7369	. . . . . . . 8 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = ( 1s /su 1s ))
32		divs1 28213	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
33	1, 32	ax-mp 5	. . . . . . . 8 ⊢ ( 1s /su 1s ) = 1s
34	31, 33	eqtrdi 2788	. . . . . . 7 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = 1s )
35	34	breq1d 5096	. . . . . 6 ⊢ (𝑛 = 1s → (( 1s /su 𝑛) ≤s 1s ↔ 1s ≤s 1s ))
36	35	rspcev 3565	. . . . 5 ⊢ (( 1s ∈ ℕs ∧ 1s ≤s 1s ) → ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
37	28, 30, 36	mp2an 693	. . . 4 ⊢ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s
38		left1s 27904	. . . . . . . 8 ⊢ ( L ‘ 1s ) = { 0s }
39		right1s 27905	. . . . . . . 8 ⊢ ( R ‘ 1s ) = ∅
40	38, 39	uneq12i 4107	. . . . . . 7 ⊢ (( L ‘ 1s ) ∪ ( R ‘ 1s )) = ({ 0s } ∪ ∅)
41		un0 4335	. . . . . . 7 ⊢ ({ 0s } ∪ ∅) = { 0s }
42	40, 41	eqtri 2760	. . . . . 6 ⊢ (( L ‘ 1s ) ∪ ( R ‘ 1s )) = { 0s }
43	42	raleqi 3294	. . . . 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 7369	. . . . . . . . . . 11 ⊢ (𝑥𝑂 = 0s → ( 1s -s 𝑥𝑂) = ( 1s -s 0s ))
46		subsid1 28077	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s -s 0s ) = 1s )
47	1, 46	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s -s 0s ) = 1s
48	45, 47	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑥𝑂 = 0s → ( 1s -s 𝑥𝑂) = 1s )
49	48	fveq2d 6839	. . . . . . . . 9 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s -s 𝑥𝑂)) = (abss‘ 1s ))
50	7, 8, 14	ltlesd 27754	. . . . . . . . . . 11 ⊢ (⊤ → 0s ≤s 1s )
51	50	mptru 1549	. . . . . . . . . 10 ⊢ 0s ≤s 1s
52		abssid 28250	. . . . . . . . . 10 ⊢ (( 1s ∈ No ∧ 0s ≤s 1s ) → (abss‘ 1s ) = 1s )
53	1, 51, 52	mp2an 693	. . . . . . . . 9 ⊢ (abss‘ 1s ) = 1s
54	49, 53	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s -s 𝑥𝑂)) = 1s )
55	54	breq2d 5098	. . . . . . 7 ⊢ (𝑥𝑂 = 0s → (( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s 1s ))
56	55	rexbidv 3162	. . . . . 6 ⊢ (𝑥𝑂 = 0s → (∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s ))
57	44, 56	ralsn 4626	. . . . 5 ⊢ (∀𝑥𝑂 ∈ { 0s }∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
58	43, 57	bitri 275	. . . 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 231	. . 3 ⊢ ∀𝑥𝑂 ∈ (( L ‘ 1s ) ∪ ( R ‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s -s 𝑥𝑂))
60	27, 59	pm3.2i 470	. 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 28504	. 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 712	1 ⊢ 1s ∈ ℝs

Syntax hints:   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∪ cun 3888  ∅c0 4274  {csn 4568   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361   No csur 27620   <s clts 27621   ≤s cles 27725   0_s c0s 27814   1_s c1s 27815   L cleft 27834   R cright 27835   +_s cadds 27968   -u_s cnegs 28028   -_s csubs 28029   /_su cdivs 28196  abs_scabss 28246  ℕ_scnns 28322  2_sc2s 28419  ℝ_screno 28498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556  ax-dc 10362

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-no 27623  df-lts 27624  df-bday 27625  df-les 27726  df-slts 27767  df-cuts 27769  df-0s 27816  df-1s 27817  df-made 27836  df-old 27837  df-left 27839  df-right 27840  df-norec 27947  df-norec2 27958  df-adds 27969  df-negs 28030  df-subs 28031  df-muls 28116  df-divs 28197  df-abss 28247  df-n0s 28323  df-nns 28324  df-2s 28420  df-reno 28499

This theorem is referenced by: (None)