Theorem 0reno 28939

Description: Surreal zero is a surreal real. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
0reno	⊢ 0s ∈ ℝs

Proof of Theorem 0reno

Dummy variables 𝑛 𝑥_𝑂 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0no 28252	. 2 ⊢ 0s ∈ No
2		1nns 28792	. . . 4 ⊢ 1s ∈ ℕs
3		0lt1s 28255	. . . . . 6 ⊢ 0s <s 1s
4		1no 28253	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1s ∈ No)
6	5	lt0negs2d 28494	. . . . . . 7 ⊢ (⊤ → ( 0s <s 1s ↔ ( -s‘ 1s ) <s 0s ))
7	6	mptru 1577	. . . . . 6 ⊢ ( 0s <s 1s ↔ ( -s‘ 1s ) <s 0s )
8	3, 7	mpbi 233	. . . . 5 ⊢ ( -s‘ 1s ) <s 0s
9	8, 3	pm3.2i 476	. . . 4 ⊢ (( -s‘ 1s ) <s 0s ∧ 0s <s 1s )
10		fveq2 6916	. . . . . . 7 ⊢ (𝑛 = 1s → ( -s‘𝑛) = ( -s‘ 1s ))
11	10	breq1d 5118	. . . . . 6 ⊢ (𝑛 = 1s → (( -s‘𝑛) <s 0s ↔ ( -s‘ 1s ) <s 0s ))
12		breq2 5112	. . . . . 6 ⊢ (𝑛 = 1s → ( 0s <s 𝑛 ↔ 0s <s 1s ))
13	11, 12	anbi12d 644	. . . . 5 ⊢ (𝑛 = 1s → ((( -s‘𝑛) <s 0s ∧ 0s <s 𝑛) ↔ (( -s‘ 1s ) <s 0s ∧ 0s <s 1s )))
14	13	rspcev 3580	. . . 4 ⊢ (( 1s ∈ ℕs ∧ (( -s‘ 1s ) <s 0s ∧ 0s <s 1s )) → ∃𝑛 ∈ ℕs (( -s‘𝑛) <s 0s ∧ 0s <s 𝑛))
15	2, 9, 14	mp2an 705	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -s‘𝑛) <s 0s ∧ 0s <s 𝑛)
16		ral0 4457	. . . 4 ⊢ ∀𝑥𝑂 ∈ ∅ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s −s 𝑥𝑂))
17		left0s 28336	. . . . . . 7 ⊢ (L‘ 0s ) = ∅
18		right0s 28337	. . . . . . 7 ⊢ (R‘ 0s ) = ∅
19	17, 18	uneq12i 4168	. . . . . 6 ⊢ ((L‘ 0s ) ∪ (R‘ 0s )) = (∅ ∪ ∅)
20		un0 4347	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
21	19, 20	eqtri 2786	. . . . 5 ⊢ ((L‘ 0s ) ∪ (R‘ 0s )) = ∅
22	21	raleqi 3321	. . . 4 ⊢ (∀𝑥𝑂 ∈ ((L‘ 0s ) ∪ (R‘ 0s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s −s 𝑥𝑂)) ↔ ∀𝑥𝑂 ∈ ∅ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s −s 𝑥𝑂)))
23	16, 22	mpbir 234	. . 3 ⊢ ∀𝑥𝑂 ∈ ((L‘ 0s ) ∪ (R‘ 0s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s −s 𝑥𝑂))
24	15, 23	pm3.2i 476	. 2 ⊢ (∃𝑛 ∈ ℕs (( -s‘𝑛) <s 0s ∧ 0s <s 𝑛) ∧ ∀𝑥𝑂 ∈ ((L‘ 0s ) ∪ (R‘ 0s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s −s 𝑥𝑂)))
25		elreno2 28938	. 2 ⊢ ( 0s ∈ ℝs ↔ ( 0s ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘𝑛) <s 0s ∧ 0s <s 𝑛) ∧ ∀𝑥𝑂 ∈ ((L‘ 0s ) ∪ (R‘ 0s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s −s 𝑥𝑂)))))
26	1, 24, 25	mpbir2an 724	1 ⊢ 0s ∈ ℝs

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ⊤wtru 1571   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ∪ cun 4152  ∅c0 4281   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  Nocsur 28054   <s clts 28055   ≤s cles 28158   0_s c0s 28248   1_s c1s 28249  Lcleft 28268  Rcright 28269   -_scnegs 28462   −_s csubs 28463   /_su cdivs 28630  abs_scabss 28680  ℕ_scnns 28756  ℝ_screno 28932

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679  ax-dc 10563

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-oadd 8518  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-1s 28251  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-muls 28550  df-divs 28631  df-abss 28681  df-n0s 28757  df-nns 28758  df-reno 28933

This theorem is used by: (None)