Theorem 0reno 28505

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 27818	. 2 ⊢ 0s ∈ No
2		1nns 28358	. . . 4 ⊢ 1s ∈ ℕs
3		0lt1s 27821	. . . . . 6 ⊢ 0s <s 1s
4		1no 27819	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (⊤ → 1s ∈ No )
6	5	lt0negs2d 28060	. . . . . . 7 ⊢ (⊤ → ( 0s <s 1s ↔ ( -us ‘ 1s ) <s 0s ))
7	6	mptru 1549	. . . . . 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 6835	. . . . . . 7 ⊢ (𝑛 = 1s → ( -us ‘𝑛) = ( -us ‘ 1s ))
11	10	breq1d 5096	. . . . . 6 ⊢ (𝑛 = 1s → (( -us ‘𝑛) <s 0s ↔ ( -us ‘ 1s ) <s 0s ))
12		breq2 5090	. . . . . 6 ⊢ (𝑛 = 1s → ( 0s <s 𝑛 ↔ 0s <s 1s ))
13	11, 12	anbi12d 633	. . . . 5 ⊢ (𝑛 = 1s → ((( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛) ↔ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )))
14	13	rspcev 3565	. . . 4 ⊢ (( 1s ∈ ℕs ∧ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )) → ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛))
15	2, 9, 14	mp2an 693	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 0s ∧ 0s <s 𝑛)
16		ral0 4439	. . . 4 ⊢ ∀𝑥𝑂 ∈ ∅ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 0s -s 𝑥𝑂))
17		left0s 27902	. . . . . . 7 ⊢ ( L ‘ 0s ) = ∅
18		right0s 27903	. . . . . . 7 ⊢ ( R ‘ 0s ) = ∅
19	17, 18	uneq12i 4107	. . . . . 6 ⊢ (( L ‘ 0s ) ∪ ( R ‘ 0s )) = (∅ ∪ ∅)
20		un0 4335	. . . . . 6 ⊢ (∅ ∪ ∅) = ∅
21	19, 20	eqtri 2760	. . . . 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 28504	. 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 712	1 ⊢ 0s ∈ ℝs

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∪ cun 3888  ∅c0 4274   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361   No csur 27620   <s clts 27621   ≤s cles 27725   0_s c0s 27814   1_s c1s 27815   L cleft 27834   R cright 27835   -u_s cnegs 28028   -_s csubs 28029   /_su cdivs 28196  abs_scabss 28246  ℕ_scnns 28322  ℝ_screno 28498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556  ax-dc 10362

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-no 27623  df-lts 27624  df-bday 27625  df-les 27726  df-slts 27767  df-cuts 27769  df-0s 27816  df-1s 27817  df-made 27836  df-old 27837  df-left 27839  df-right 27840  df-norec 27947  df-norec2 27958  df-adds 27969  df-negs 28030  df-subs 28031  df-muls 28116  df-divs 28197  df-abss 28247  df-n0s 28323  df-nns 28324  df-reno 28499

This theorem is referenced by: (None)