Theorem 0reno 28654

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 27967	. 2 ⊢ 0s ∈ No
2		1nns 28507	. . . 4 ⊢ 1s ∈ ℕs
3		0lt1s 27970	. . . . . 6 ⊢ 0s <s 1s
4		1no 27968	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1s ∈ No )
6	5	lt0negs2d 28209	. . . . . . 7 ⊢ (⊤ → ( 0s <s 1s ↔ ( -us ‘ 1s ) <s 0s ))
7	6	mptru 1574	. . . . . 6 ⊢ ( 0s <s 1s ↔ ( -us ‘ 1s ) <s 0s )
8	3, 7	mpbi 233	. . . . 5 ⊢ ( -us ‘ 1s ) <s 0s
9	8, 3	pm3.2i 475	. . . 4 ⊢ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )
10		fveq2 6882	. . . . . . 7 ⊢ (𝑛 = 1s → ( -us ‘𝑛) = ( -us ‘ 1s ))
11	10	breq1d 5123	. . . . . 6 ⊢ (𝑛 = 1s → (( -us ‘𝑛) <s 0s ↔ ( -us ‘ 1s ) <s 0s ))
12		breq2 5117	. . . . . 6 ⊢ (𝑛 = 1s → ( 0s <s 𝑛 ↔ 0s <s 1s ))
13	11, 12	anbi12d 643	. . . . 5 ⊢ (𝑛 = 1s → ((( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛) ↔ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )))
14	13	rspcev 3590	. . . 4 ⊢ (( 1s ∈ ℕs ∧ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛))
15	2, 9, 14	mp2an 704	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛)
16		ral0 4464	. . . 4 ⊢ ∀𝑥𝑂 ∈ ∅ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s -s 𝑥𝑂))
17		left0s 28051	. . . . . . 7 ⊢ ( L ‘ 0s ) = ∅
18		right0s 28052	. . . . . . 7 ⊢ ( R ‘ 0s ) = ∅
19	17, 18	uneq12i 4128	. . . . . 6 ⊢ (( L ‘ 0s ) ∪ ( R ‘ 0s )) = (∅ ∪ ∅)
20		un0 4358	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
21	19, 20	eqtri 2792	. . . . 5 ⊢ (( L ‘ 0s ) ∪ ( R ‘ 0s )) = ∅
22	21	raleqi 3327	. . . 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 475	. 2 ⊢ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛) ∧ ∀𝑥𝑂 ∈ (( L ‘ 0s ) ∪ ( R ‘ 0s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s -s 𝑥𝑂)))
25		elreno2 28653	. 2 ⊢ ( 0s ∈ ℝs ↔ ( 0s ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘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 723	1 ⊢ 0s ∈ ℝs

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ⊤wtru 1568   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ∪ cun 3911  ∅c0 4294   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411   No csur 27769   <s clts 27770   ≤s cles 27873   0_s c0s 27963   1_s c1s 27964   L cleft 27983   R cright 27984   -u_s cnegs 28177   -_s csubs 28178   /_su cdivs 28345  abs_scabss 28395  ℕ_scnns 28471  ℝ_screno 28647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9609  ax-dc 10429

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-nadd 8651  df-no 27772  df-lts 27773  df-bday 27774  df-les 27874  df-slts 27916  df-cuts 27918  df-0s 27965  df-1s 27966  df-made 27985  df-old 27986  df-left 27988  df-right 27989  df-norec 28096  df-norec2 28107  df-adds 28118  df-negs 28179  df-subs 28180  df-muls 28265  df-divs 28346  df-abss 28396  df-n0s 28472  df-nns 28473  df-reno 28648

This theorem is referenced by: (None)