Theorem n0sbday 28269

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 6896	. . 3 ⊢ (𝑚 = 0s → ( bday ‘𝑚) = ( bday ‘ 0s ))
2	1	eleq1d 2810	. 2 ⊢ (𝑚 = 0s → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘ 0s ) ∈ ω))
3		fveq2 6896	. . 3 ⊢ (𝑚 = 𝑛 → ( bday ‘𝑚) = ( bday ‘𝑛))
4	3	eleq1d 2810	. 2 ⊢ (𝑚 = 𝑛 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝑛) ∈ ω))
5		fveq2 6896	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ( bday ‘𝑚) = ( bday ‘(𝑛 +s 1s )))
6	5	eleq1d 2810	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘(𝑛 +s 1s )) ∈ ω))
7		fveq2 6896	. . 3 ⊢ (𝑚 = 𝐴 → ( bday ‘𝑚) = ( bday ‘𝐴))
8	7	eleq1d 2810	. 2 ⊢ (𝑚 = 𝐴 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝐴) ∈ ω))
9		bday0s 27807	. . 3 ⊢ ( bday ‘ 0s ) = ∅
10		peano1 7895	. . 3 ⊢ ∅ ∈ ω
11	9, 10	eqeltri 2821	. 2 ⊢ ( bday ‘ 0s ) ∈ ω
12		peano2n0s 28252	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
13		n0scut 28255	. . . . . . . . 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 28245	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0s → 𝑛 ∈ No )
16		1sno 27806	. . . . . . . . . . 11 ⊢ 1s ∈ No
17		pncans 28028	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ No ∧ 1s ∈ No ) → ((𝑛 +s 1s ) -s 1s ) = 𝑛)
18	15, 16, 17	sylancl 584	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → ((𝑛 +s 1s ) -s 1s ) = 𝑛)
19	18	sneqd 4642	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → {((𝑛 +s 1s ) -s 1s )} = {𝑛})
20	19	oveq1d 7434	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ({((𝑛 +s 1s ) -s 1s )} |s ∅) = ({𝑛} |s ∅))
21	14, 20	eqtrd 2765	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({𝑛} |s ∅))
22	21	fveq2d 6900	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) = ( bday ‘({𝑛} |s ∅)))
23		snelpwi 5445	. . . . . . . 8 ⊢ (𝑛 ∈ No → {𝑛} ∈ 𝒫 No )
24		nulssgt 27777	. . . . . . . 8 ⊢ ({𝑛} ∈ 𝒫 No → {𝑛} <<s ∅)
25	15, 23, 24	3syl 18	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → {𝑛} <<s ∅)
26		un0 4392	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
27	26	imaeq2i 6062	. . . . . . . . 9 ⊢ ( bday “ ({𝑛} ∪ ∅)) = ( bday “ {𝑛})
28		bdayfn 27752	. . . . . . . . . 10 ⊢ bday Fn No
29		fnsnfv 6976	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑛 ∈ No ) → {( bday ‘𝑛)} = ( bday “ {𝑛}))
30	28, 15, 29	sylancr 585	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → {( bday ‘𝑛)} = ( bday “ {𝑛}))
31	27, 30	eqtr4id 2784	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) = {( bday ‘𝑛)})
32		fvex 6909	. . . . . . . . . 10 ⊢ ( bday ‘𝑛) ∈ V
33	32	sucid 6453	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ suc ( bday ‘𝑛)
34		snssi 4813	. . . . . . . . 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 4031	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛))
37		bdayelon 27755	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ On
38	37	onsuci 7843	. . . . . . . 8 ⊢ suc ( bday ‘𝑛) ∈ On
39		scutbdaybnd 27794	. . . . . . . 8 ⊢ (({𝑛} <<s ∅ ∧ suc ( bday ‘𝑛) ∈ On ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
40	38, 39	mp3an2 1445	. . . . . . 7 ⊢ (({𝑛} <<s ∅ ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
41	25, 36, 40	syl2anc 582	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
42	22, 41	eqsstrd 4015	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛))
43		bdayelon 27755	. . . . . 6 ⊢ ( bday ‘(𝑛 +s 1s )) ∈ On
44		onsssuc 6461	. . . . . 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 690	. . . . 5 ⊢ (( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛) ↔ ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
46	42, 45	sylib 217	. . . 4 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
47		peano2 7897	. . . . 5 ⊢ (( bday ‘𝑛) ∈ ω → suc ( bday ‘𝑛) ∈ ω)
48		peano2 7897	. . . . 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 7882	. . . 4 ⊢ ((( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛) ∧ suc suc ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
51	46, 49, 50	syl2an 594	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
52	51	ex 411	. 2 ⊢ (𝑛 ∈ ℕ0s → (( bday ‘𝑛) ∈ ω → ( bday ‘(𝑛 +s 1s )) ∈ ω))
53	2, 4, 6, 8, 11, 52	n0sind 28254	1 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098   ∪ cun 3942   ⊆ wss 3944  ∅c0 4322  𝒫 cpw 4604  {csn 4630   class class class wbr 5149   “ cima 5681  Oncon0 6371  suc csuc 6373   Fn wfn 6544  ‘cfv 6549  (class class class)co 7419  ωcom 7871   No csur 27618   bday cbday 27620   <<s csslt 27759   |s cscut 27761   0_s c0s 27801   1_s c1s 27802   +_s cadds 27922   -_s csubs 27979  ℕ_0scnn0s 28235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-nadd 8687  df-no 27621  df-slt 27622  df-bday 27623  df-sle 27724  df-sslt 27760  df-scut 27762  df-0s 27803  df-1s 27804  df-made 27820  df-old 27821  df-left 27823  df-right 27824  df-norec 27901  df-norec2 27912  df-adds 27923  df-negs 27980  df-subs 27981  df-n0s 28237

This theorem is referenced by:  n0ssold  28270  zsbday  28292