Theorem 0reno 28492

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 27805	. 2 ⊢ 0s ∈ No
2		1nns 28345	. . . 4 ⊢ 1s ∈ ℕs
3		0lt1s 27808	. . . . . 6 ⊢ 0s <s 1s
4		1no 27806	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1s ∈ No )
6	5	lt0negs2d 28047	. . . . . . 7 ⊢ (⊤ → ( 0s <s 1s ↔ ( -us ‘ 1s ) <s 0s ))
7	6	mptru 1548	. . . . . 6 ⊢ ( 0s <s 1s ↔ ( -us ‘ 1s ) <s 0s )
8	3, 7	mpbi 230	. . . . 5 ⊢ ( -us ‘ 1s ) <s 0s
9	8, 3	pm3.2i 470	. . . 4 ⊢ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )
10		fveq2 6834	. . . . . . 7 ⊢ (𝑛 = 1s → ( -us ‘𝑛) = ( -us ‘ 1s ))
11	10	breq1d 5108	. . . . . 6 ⊢ (𝑛 = 1s → (( -us ‘𝑛) <s 0s ↔ ( -us ‘ 1s ) <s 0s ))
12		breq2 5102	. . . . . 6 ⊢ (𝑛 = 1s → ( 0s <s 𝑛 ↔ 0s <s 1s ))
13	11, 12	anbi12d 632	. . . . 5 ⊢ (𝑛 = 1s → ((( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛) ↔ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )))
14	13	rspcev 3576	. . . 4 ⊢ (( 1s ∈ ℕs ∧ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛))
15	2, 9, 14	mp2an 692	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛)
16		ral0 4451	. . . 4 ⊢ ∀𝑥𝑂 ∈ ∅ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s -s 𝑥𝑂))
17		left0s 27889	. . . . . . 7 ⊢ ( L ‘ 0s ) = ∅
18		right0s 27890	. . . . . . 7 ⊢ ( R ‘ 0s ) = ∅
19	17, 18	uneq12i 4118	. . . . . 6 ⊢ (( L ‘ 0s ) ∪ ( R ‘ 0s )) = (∅ ∪ ∅)
20		un0 4346	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
21	19, 20	eqtri 2759	. . . . 5 ⊢ (( L ‘ 0s ) ∪ ( R ‘ 0s )) = ∅
22	21	raleqi 3294	. . . 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 231	. . 3 ⊢ ∀𝑥𝑂 ∈ (( L ‘ 0s ) ∪ ( R ‘ 0s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s -s 𝑥𝑂))
24	15, 23	pm3.2i 470	. 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 28491	. 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 711	1 ⊢ 0s ∈ ℝs

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ⊤wtru 1542   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ∪ cun 3899  ∅c0 4285   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358   No csur 27607   <s clts 27608   ≤s cles 27712   0_s c0s 27801   1_s c1s 27802   L cleft 27821   R cright 27822   -u_s cnegs 28015   -_s csubs 28016   /_su cdivs 28183  abs_scabss 28233  ℕ_scnns 28309  ℝ_screno 28485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550  ax-dc 10356

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27934  df-norec2 27945  df-adds 27956  df-negs 28017  df-subs 28018  df-muls 28103  df-divs 28184  df-abss 28234  df-n0s 28310  df-nns 28311  df-reno 28486

This theorem is referenced by: (None)