Theorem n0sbday 28355

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 6905	. . 3 ⊢ (𝑚 = 0s → ( bday ‘𝑚) = ( bday ‘ 0s ))
2	1	eleq1d 2825	. 2 ⊢ (𝑚 = 0s → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘ 0s ) ∈ ω))
3		fveq2 6905	. . 3 ⊢ (𝑚 = 𝑛 → ( bday ‘𝑚) = ( bday ‘𝑛))
4	3	eleq1d 2825	. 2 ⊢ (𝑚 = 𝑛 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝑛) ∈ ω))
5		fveq2 6905	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ( bday ‘𝑚) = ( bday ‘(𝑛 +s 1s )))
6	5	eleq1d 2825	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘(𝑛 +s 1s )) ∈ ω))
7		fveq2 6905	. . 3 ⊢ (𝑚 = 𝐴 → ( bday ‘𝑚) = ( bday ‘𝐴))
8	7	eleq1d 2825	. 2 ⊢ (𝑚 = 𝐴 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝐴) ∈ ω))
9		bday0s 27874	. . 3 ⊢ ( bday ‘ 0s ) = ∅
10		peano1 7911	. . 3 ⊢ ∅ ∈ ω
11	9, 10	eqeltri 2836	. 2 ⊢ ( bday ‘ 0s ) ∈ ω
12		peano2n0s 28336	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
13		n0scut 28339	. . . . . . . . 9 ⊢ ((𝑛 +s 1s ) ∈ ℕ0s → (𝑛 +s 1s ) = ({((𝑛 +s 1s ) -s 1s )} |s ∅))
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({((𝑛 +s 1s ) -s 1s )} |s ∅))
15		n0sno 28329	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0s → 𝑛 ∈ No )
16		1sno 27873	. . . . . . . . . . 11 ⊢ 1s ∈ No
17		pncans 28103	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ No ∧ 1s ∈ No ) → ((𝑛 +s 1s ) -s 1s ) = 𝑛)
18	15, 16, 17	sylancl 586	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → ((𝑛 +s 1s ) -s 1s ) = 𝑛)
19	18	sneqd 4637	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → {((𝑛 +s 1s ) -s 1s )} = {𝑛})
20	19	oveq1d 7447	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ({((𝑛 +s 1s ) -s 1s )} |s ∅) = ({𝑛} |s ∅))
21	14, 20	eqtrd 2776	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({𝑛} |s ∅))
22	21	fveq2d 6909	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) = ( bday ‘({𝑛} |s ∅)))
23		snelpwi 5447	. . . . . . . 8 ⊢ (𝑛 ∈ No → {𝑛} ∈ 𝒫 No )
24		nulssgt 27844	. . . . . . . 8 ⊢ ({𝑛} ∈ 𝒫 No → {𝑛} <<s ∅)
25	15, 23, 24	3syl 18	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → {𝑛} <<s ∅)
26		un0 4393	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
27	26	imaeq2i 6075	. . . . . . . . 9 ⊢ ( bday “ ({𝑛} ∪ ∅)) = ( bday “ {𝑛})
28		bdayfn 27819	. . . . . . . . . 10 ⊢ bday Fn No
29		fnsnfv 6987	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑛 ∈ No ) → {( bday ‘𝑛)} = ( bday “ {𝑛}))
30	28, 15, 29	sylancr 587	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → {( bday ‘𝑛)} = ( bday “ {𝑛}))
31	27, 30	eqtr4id 2795	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) = {( bday ‘𝑛)})
32		fvex 6918	. . . . . . . . . 10 ⊢ ( bday ‘𝑛) ∈ V
33	32	sucid 6465	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ suc ( bday ‘𝑛)
34		snssi 4807	. . . . . . . . 9 ⊢ (( bday ‘𝑛) ∈ suc ( bday ‘𝑛) → {( bday ‘𝑛)} ⊆ suc ( bday ‘𝑛))
35	33, 34	ax-mp 5	. . . . . . . 8 ⊢ {( bday ‘𝑛)} ⊆ suc ( bday ‘𝑛)
36	31, 35	eqsstrdi 4027	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛))
37		bdayelon 27822	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ On
38	37	onsuci 7860	. . . . . . . 8 ⊢ suc ( bday ‘𝑛) ∈ On
39		scutbdaybnd 27861	. . . . . . . 8 ⊢ (({𝑛} <<s ∅ ∧ suc ( bday ‘𝑛) ∈ On ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
40	38, 39	mp3an2 1450	. . . . . . 7 ⊢ (({𝑛} <<s ∅ ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
41	25, 36, 40	syl2anc 584	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
42	22, 41	eqsstrd 4017	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛))
43		bdayelon 27822	. . . . . 6 ⊢ ( bday ‘(𝑛 +s 1s )) ∈ On
44		onsssuc 6473	. . . . . 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)))
45	43, 38, 44	mp2an 692	. . . . 5 ⊢ (( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛) ↔ ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
46	42, 45	sylib 218	. . . 4 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
47		peano2 7913	. . . . 5 ⊢ (( bday ‘𝑛) ∈ ω → suc ( bday ‘𝑛) ∈ ω)
48		peano2 7913	. . . . 5 ⊢ (suc ( bday ‘𝑛) ∈ ω → suc suc ( bday ‘𝑛) ∈ ω)
49	47, 48	syl 17	. . . 4 ⊢ (( bday ‘𝑛) ∈ ω → suc suc ( bday ‘𝑛) ∈ ω)
50		elnn 7899	. . . 4 ⊢ ((( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛) ∧ suc suc ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
51	46, 49, 50	syl2an 596	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
52	51	ex 412	. 2 ⊢ (𝑛 ∈ ℕ0s → (( bday ‘𝑛) ∈ ω → ( bday ‘(𝑛 +s 1s )) ∈ ω))
53	2, 4, 6, 8, 11, 52	n0sind 28338	1 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2107   ∪ cun 3948   ⊆ wss 3950  ∅c0 4332  𝒫 cpw 4599  {csn 4625   class class class wbr 5142   “ cima 5687  Oncon0 6383  suc csuc 6385   Fn wfn 6555  ‘cfv 6560  (class class class)co 7432  ωcom 7888   No csur 27685   bday cbday 27687   <<s csslt 27826   |s cscut 27828   0_s c0s 27868   1_s c1s 27869   +_s cadds 27993   -_s csubs 28053  ℕ_0scnn0s 28319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-1s 27871  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054  df-subs 28055  df-n0s 28321

This theorem is referenced by:  n0ssold  28356  zsbday  28393