Theorem n0sbday 28250

Description: A non-negative surreal integer has a finite birthday. (Contributed by Scott Fenton, 18-Apr-2025.)

Assertion

Ref	Expression
n0sbday	⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)

Proof of Theorem n0sbday

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6860	. . 3 ⊢ (𝑚 = 0s → ( bday ‘𝑚) = ( bday ‘ 0s ))
2	1	eleq1d 2814	. 2 ⊢ (𝑚 = 0s → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘ 0s ) ∈ ω))
3		fveq2 6860	. . 3 ⊢ (𝑚 = 𝑛 → ( bday ‘𝑚) = ( bday ‘𝑛))
4	3	eleq1d 2814	. 2 ⊢ (𝑚 = 𝑛 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝑛) ∈ ω))
5		fveq2 6860	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ( bday ‘𝑚) = ( bday ‘(𝑛 +s 1s )))
6	5	eleq1d 2814	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘(𝑛 +s 1s )) ∈ ω))
7		fveq2 6860	. . 3 ⊢ (𝑚 = 𝐴 → ( bday ‘𝑚) = ( bday ‘𝐴))
8	7	eleq1d 2814	. 2 ⊢ (𝑚 = 𝐴 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝐴) ∈ ω))
9		bday0s 27746	. . 3 ⊢ ( bday ‘ 0s ) = ∅
10		peano1 7867	. . 3 ⊢ ∅ ∈ ω
11	9, 10	eqeltri 2825	. 2 ⊢ ( bday ‘ 0s ) ∈ ω
12		n0scut2 28233	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({𝑛} |s ∅))
13	12	fveq2d 6864	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) = ( bday ‘({𝑛} |s ∅)))
14		n0sno 28222	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 𝑛 ∈ No )
15		snelpwi 5405	. . . . . . . 8 ⊢ (𝑛 ∈ No → {𝑛} ∈ 𝒫 No )
16		nulssgt 27716	. . . . . . . 8 ⊢ ({𝑛} ∈ 𝒫 No → {𝑛} <<s ∅)
17	14, 15, 16	3syl 18	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → {𝑛} <<s ∅)
18		un0 4359	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
19	18	imaeq2i 6031	. . . . . . . . 9 ⊢ ( bday “ ({𝑛} ∪ ∅)) = ( bday “ {𝑛})
20		bdayfn 27691	. . . . . . . . . 10 ⊢ bday Fn No
21		fnsnfv 6942	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑛 ∈ No ) → {( bday ‘𝑛)} = ( bday “ {𝑛}))
22	20, 14, 21	sylancr 587	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → {( bday ‘𝑛)} = ( bday “ {𝑛}))
23	19, 22	eqtr4id 2784	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) = {( bday ‘𝑛)})
24		fvex 6873	. . . . . . . . . 10 ⊢ ( bday ‘𝑛) ∈ V
25	24	sucid 6418	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ suc ( bday ‘𝑛)
26		snssi 4774	. . . . . . . . 9 ⊢ (( bday ‘𝑛) ∈ suc ( bday ‘𝑛) → {( bday ‘𝑛)} ⊆ suc ( bday ‘𝑛))
27	25, 26	ax-mp 5	. . . . . . . 8 ⊢ {( bday ‘𝑛)} ⊆ suc ( bday ‘𝑛)
28	23, 27	eqsstrdi 3993	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛))
29		bdayelon 27694	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ On
30	29	onsuci 7816	. . . . . . . 8 ⊢ suc ( bday ‘𝑛) ∈ On
31		scutbdaybnd 27733	. . . . . . . 8 ⊢ (({𝑛} <<s ∅ ∧ suc ( bday ‘𝑛) ∈ On ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
32	30, 31	mp3an2 1451	. . . . . . 7 ⊢ (({𝑛} <<s ∅ ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
33	17, 28, 32	syl2anc 584	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
34	13, 33	eqsstrd 3983	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛))
35		bdayelon 27694	. . . . . 6 ⊢ ( bday ‘(𝑛 +s 1s )) ∈ On
36		onsssuc 6426	. . . . . 6 ⊢ ((( bday ‘(𝑛 +s 1s )) ∈ On ∧ suc ( bday ‘𝑛) ∈ On) → (( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛) ↔ ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛)))
37	35, 30, 36	mp2an 692	. . . . 5 ⊢ (( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛) ↔ ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
38	34, 37	sylib 218	. . . 4 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
39		peano2 7868	. . . . 5 ⊢ (( bday ‘𝑛) ∈ ω → suc ( bday ‘𝑛) ∈ ω)
40		peano2 7868	. . . . 5 ⊢ (suc ( bday ‘𝑛) ∈ ω → suc suc ( bday ‘𝑛) ∈ ω)
41	39, 40	syl 17	. . . 4 ⊢ (( bday ‘𝑛) ∈ ω → suc suc ( bday ‘𝑛) ∈ ω)
42		elnn 7855	. . . 4 ⊢ ((( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛) ∧ suc suc ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
43	38, 41, 42	syl2an 596	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
44	43	ex 412	. 2 ⊢ (𝑛 ∈ ℕ0s → (( bday ‘𝑛) ∈ ω → ( bday ‘(𝑛 +s 1s )) ∈ ω))
45	2, 4, 6, 8, 11, 44	n0sind 28231	1 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ∪ cun 3914   ⊆ wss 3916  ∅c0 4298  𝒫 cpw 4565  {csn 4591   class class class wbr 5109   “ cima 5643  Oncon0 6334  suc csuc 6336   Fn wfn 6508  ‘cfv 6513  (class class class)co 7389  ωcom 7844   No csur 27557   bday cbday 27559   <<s csslt 27698   |s cscut 27700   0_s c0s 27740   1_s c1s 27741   +_s cadds 27872  ℕ_0scnn0s 28212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-2o 8437  df-nadd 8632  df-no 27560  df-slt 27561  df-bday 27562  df-sle 27663  df-sslt 27699  df-scut 27701  df-0s 27742  df-1s 27743  df-made 27761  df-old 27762  df-left 27764  df-right 27765  df-norec 27851  df-norec2 27862  df-adds 27873  df-negs 27933  df-subs 27934  df-n0s 28214

This theorem is referenced by:  n0ssold  28251  onltn0s  28254  bdayn0sf1o  28265  zsbday  28300