Theorem 0reno 28586

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 27899	. 2 ⊢ 0s ∈ No
2		1nns 28439	. . . 4 ⊢ 1s ∈ ℕs
3		0lt1s 27902	. . . . . 6 ⊢ 0s <s 1s
4		1no 27900	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1s ∈ No )
6	5	lt0negs2d 28141	. . . . . . 7 ⊢ (⊤ → ( 0s <s 1s ↔ ( -us ‘ 1s ) <s 0s ))
7	6	mptru 1567	. . . . . 6 ⊢ ( 0s <s 1s ↔ ( -us ‘ 1s ) <s 0s )
8	3, 7	mpbi 232	. . . . 5 ⊢ ( -us ‘ 1s ) <s 0s
9	8, 3	pm3.2i 474	. . . 4 ⊢ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )
10		fveq2 6867	. . . . . . 7 ⊢ (𝑛 = 1s → ( -us ‘𝑛) = ( -us ‘ 1s ))
11	10	breq1d 5110	. . . . . 6 ⊢ (𝑛 = 1s → (( -us ‘𝑛) <s 0s ↔ ( -us ‘ 1s ) <s 0s ))
12		breq2 5104	. . . . . 6 ⊢ (𝑛 = 1s → ( 0s <s 𝑛 ↔ 0s <s 1s ))
13	11, 12	anbi12d 641	. . . . 5 ⊢ (𝑛 = 1s → ((( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛) ↔ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )))
14	13	rspcev 3581	. . . 4 ⊢ (( 1s ∈ ℕs ∧ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛))
15	2, 9, 14	mp2an 702	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛)
16		ral0 4452	. . . 4 ⊢ ∀𝑥𝑂 ∈ ∅ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s -s 𝑥𝑂))
17		left0s 27983	. . . . . . 7 ⊢ ( L ‘ 0s ) = ∅
18		right0s 27984	. . . . . . 7 ⊢ ( R ‘ 0s ) = ∅
19	17, 18	uneq12i 4119	. . . . . 6 ⊢ (( L ‘ 0s ) ∪ ( R ‘ 0s )) = (∅ ∪ ∅)
20		un0 4348	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
21	19, 20	eqtri 2785	. . . . 5 ⊢ (( L ‘ 0s ) ∪ ( R ‘ 0s )) = ∅
22	21	raleqi 3318	. . . 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 233	. . 3 ⊢ ∀𝑥𝑂 ∈ (( L ‘ 0s ) ∪ ( R ‘ 0s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s -s 𝑥𝑂))
24	15, 23	pm3.2i 474	. 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 28585	. 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 721	1 ⊢ 0s ∈ ℝs

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560  ⊤wtru 1561   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ∪ cun 3902  ∅c0 4285   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396   No csur 27701   <s clts 27702   ≤s cles 27805   0_s c0s 27895   1_s c1s 27896   L cleft 27915   R cright 27916   -u_s cnegs 28109   -_s csubs 28110   /_su cdivs 28277  abs_scabss 28327  ℕ_scnns 28403  ℝ_screno 28579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-dc 10403

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-nadd 8636  df-no 27704  df-lts 27705  df-bday 27706  df-les 27806  df-slts 27848  df-cuts 27850  df-0s 27897  df-1s 27898  df-made 27917  df-old 27918  df-left 27920  df-right 27921  df-norec 28028  df-norec2 28039  df-adds 28050  df-negs 28111  df-subs 28112  df-muls 28197  df-divs 28278  df-abss 28328  df-n0s 28404  df-nns 28405  df-reno 28580

This theorem is referenced by: (None)