Theorem n0bday 28720

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

Assertion

Ref	Expression
n0bday	⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)

Proof of Theorem n0bday

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

Step	Hyp	Ref	Expression
1		fveq2 6877	. . 3 ⊢ (𝑚 = 0s → ( bday ‘𝑚) = ( bday ‘ 0s ))
2	1	eleq1d 2846	. 2 ⊢ (𝑚 = 0s → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘ 0s ) ∈ ω))
3		fveq2 6877	. . 3 ⊢ (𝑚 = 𝑛 → ( bday ‘𝑚) = ( bday ‘𝑛))
4	3	eleq1d 2846	. 2 ⊢ (𝑚 = 𝑛 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝑛) ∈ ω))
5		fveq2 6877	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ( bday ‘𝑚) = ( bday ‘(𝑛 +s 1s )))
6	5	eleq1d 2846	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘(𝑛 +s 1s )) ∈ ω))
7		fveq2 6877	. . 3 ⊢ (𝑚 = 𝐴 → ( bday ‘𝑚) = ( bday ‘𝐴))
8	7	eleq1d 2846	. 2 ⊢ (𝑚 = 𝐴 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝐴) ∈ ω))
9		bday0 28179	. . 3 ⊢ ( bday ‘ 0s ) = ∅
10		peano1 7889	. . 3 ⊢ ∅ ∈ ω
11	9, 10	eqeltri 2857	. 2 ⊢ ( bday ‘ 0s ) ∈ ω
12		n0cut2 28703	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({𝑛} |s ∅))
13	12	fveq2d 6881	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) = ( bday ‘({𝑛} |s ∅)))
14		n0no 28691	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 𝑛 ∈ No )
15		snelpwi 5412	. . . . . . . 8 ⊢ (𝑛 ∈ No → {𝑛} ∈ 𝒫 No )
16		nulsgts 28144	. . . . . . . 8 ⊢ ({𝑛} ∈ 𝒫 No → {𝑛} <<s ∅)
17	14, 15, 16	3syl 19	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → {𝑛} <<s ∅)
18		un0 4344	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
19	18	imaeq2i 6052	. . . . . . . . 9 ⊢ ( bday “ ({𝑛} ∪ ∅)) = ( bday “ {𝑛})
20		bdayfn 28116	. . . . . . . . . 10 ⊢ bday Fn No
21		fnsnfv 6956	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑛 ∈ No ) → {( bday ‘𝑛)} = ( bday “ {𝑛}))
22	20, 14, 21	sylancr 599	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → {( bday ‘𝑛)} = ( bday “ {𝑛}))
23	19, 22	eqtr4id 2815	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) = {( bday ‘𝑛)})
24		fvex 6890	. . . . . . . . . 10 ⊢ ( bday ‘𝑛) ∈ V
25	24	sucid 6440	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ suc ( bday ‘𝑛)
26		snssi 4746	. . . . . . . . 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 3975	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛))
29		bdayon 28120	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ On
30	29	onsuci 7839	. . . . . . . 8 ⊢ suc ( bday ‘𝑛) ∈ On
31		cutbdaybnd 28163	. . . . . . . 8 ⊢ (({𝑛} <<s ∅ ∧ suc ( bday ‘𝑛) ∈ On ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
32	30, 31	mp3an2 1478	. . . . . . 7 ⊢ (({𝑛} <<s ∅ ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
33	17, 28, 32	syl2anc 596	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
34	13, 33	eqsstrd 3965	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛))
35		bdayon 28120	. . . . . 6 ⊢ ( bday ‘(𝑛 +s 1s )) ∈ On
36		onsssuc 6448	. . . . . 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 705	. . . . 5 ⊢ (( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛) ↔ ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
38	34, 37	sylib 221	. . . 4 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
39		peano2 7890	. . . . 5 ⊢ (( bday ‘𝑛) ∈ ω → suc ( bday ‘𝑛) ∈ ω)
40		peano2 7890	. . . . 5 ⊢ (suc ( bday ‘𝑛) ∈ ω → suc suc ( bday ‘𝑛) ∈ ω)
41	39, 40	syl 18	. . . 4 ⊢ (( bday ‘𝑛) ∈ ω → suc suc ( bday ‘𝑛) ∈ ω)
42		elnn 7877	. . . 4 ⊢ ((( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛) ∧ suc suc ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
43	38, 41, 42	syl2an 608	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
44	43	ex 418	. 2 ⊢ (𝑛 ∈ ℕ0s → (( bday ‘𝑛) ∈ ω → ( bday ‘(𝑛 +s 1s )) ∈ ω))
45	2, 4, 6, 8, 11, 44	n0sind 28701	1 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584   class class class wbr 5103   “ cima 5654  Oncon0 6355  suc csuc 6357   Fn wfn 6526  ‘cfv 6531  (class class class)co 7412  ωcom 7866   No csur 27979   bday cbday 27981   <<s cslts 28125   |s ccuts 28127   0_s c0s 28173   1_s c1s 28174   +_s cadds 28327  ℕ_0scn0s 28680

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-nadd 8659  df-no 27982  df-lts 27983  df-bday 27984  df-les 28084  df-slts 28126  df-cuts 28128  df-0s 28175  df-1s 28176  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec 28306  df-norec2 28317  df-adds 28328  df-negs 28389  df-subs 28390  df-n0s 28682

This theorem is used by:  n0ssoldg  28721  eln0s2  28725  onltn0s  28726  bdayn0sf1o  28738  zsbday  28774  bdayfinbndlem1  28835  z12bdaylem  28852