Theorem 0reno 28891

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 28204	. 2 ⊢ 0s ∈ No
2		1nns 28744	. . . 4 ⊢ 1s ∈ ℕs
3		0lt1s 28207	. . . . . 6 ⊢ 0s <s 1s
4		1no 28205	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1s ∈ No)
6	5	lt0negs2d 28446	. . . . . . 7 ⊢ (⊤ → ( 0s <s 1s ↔ ( -s‘ 1s ) <s 0s ))
7	6	mptru 1576	. . . . . 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 6890	. . . . . . 7 ⊢ (𝑛 = 1s → ( -s‘𝑛) = ( -s‘ 1s ))
11	10	breq1d 5112	. . . . . 6 ⊢ (𝑛 = 1s → (( -s‘𝑛) <s 0s ↔ ( -s‘ 1s ) <s 0s ))
12		breq2 5106	. . . . . 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 3576	. . . 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 4453	. . . 4 ⊢ ∀𝑥𝑂 ∈ ∅ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s −s 𝑥𝑂))
17		left0s 28288	. . . . . . 7 ⊢ (L‘ 0s ) = ∅
18		right0s 28289	. . . . . . 7 ⊢ (R‘ 0s ) = ∅
19	17, 18	uneq12i 4164	. . . . . 6 ⊢ ((L‘ 0s ) ∪ (R‘ 0s )) = (∅ ∪ ∅)
20		un0 4343	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
21	19, 20	eqtri 2783	. . . . 5 ⊢ ((L‘ 0s ) ∪ (R‘ 0s )) = ∅
22	21	raleqi 3317	. . . 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 28890	. 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 1569  ⊤wtru 1570   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ∪ cun 4148  ∅c0 4277   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Nocsur 28006   <s clts 28007   ≤s cles 28110   0_s c0s 28200   1_s c1s 28201  Lcleft 28220  Rcright 28221   -_scnegs 28414   −_s csubs 28415   /_su cdivs 28582  abs_scabss 28632  ℕ_scnns 28708  ℝ_screno 28884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-dc 10531

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-2o 8485  df-oadd 8488  df-nadd 8683  df-no 28009  df-lts 28010  df-bday 28011  df-les 28111  df-slts 28153  df-cuts 28155  df-0s 28202  df-1s 28203  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec 28333  df-norec2 28344  df-adds 28355  df-negs 28416  df-subs 28417  df-muls 28502  df-divs 28583  df-abss 28633  df-n0s 28709  df-nns 28710  df-reno 28885

This theorem is used by: (None)