Theorem n0bday 28743

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 6885	. . 3 ⊢ (𝑚 = 0s → (bday‘𝑚) = (bday‘ 0s ))
2	1	eleq1d 2845	. 2 ⊢ (𝑚 = 0s → ((bday‘𝑚) ∈ ω ↔ (bday‘ 0s ) ∈ ω))
3		fveq2 6885	. . 3 ⊢ (𝑚 = 𝑛 → (bday‘𝑚) = (bday‘𝑛))
4	3	eleq1d 2845	. 2 ⊢ (𝑚 = 𝑛 → ((bday‘𝑚) ∈ ω ↔ (bday‘𝑛) ∈ ω))
5		fveq2 6885	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (bday‘𝑚) = (bday‘(𝑛 +s 1s )))
6	5	eleq1d 2845	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → ((bday‘𝑚) ∈ ω ↔ (bday‘(𝑛 +s 1s )) ∈ ω))
7		fveq2 6885	. . 3 ⊢ (𝑚 = 𝐴 → (bday‘𝑚) = (bday‘𝐴))
8	7	eleq1d 2845	. 2 ⊢ (𝑚 = 𝐴 → ((bday‘𝑚) ∈ ω ↔ (bday‘𝐴) ∈ ω))
9		bday0 28202	. . 3 ⊢ (bday‘ 0s ) = ∅
10		peano1 7900	. . 3 ⊢ ∅ ∈ ω
11	9, 10	eqeltri 2856	. 2 ⊢ (bday‘ 0s ) ∈ ω
12		n0cut2 28726	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({𝑛} |s ∅))
13	12	fveq2d 6889	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → (bday‘(𝑛 +s 1s )) = (bday‘({𝑛} |s ∅)))
14		n0no 28714	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 𝑛 ∈ No)
15		snelpwi 5411	. . . . . . . 8 ⊢ (𝑛 ∈ No → {𝑛} ∈ 𝒫 No)
16		nulsgts 28167	. . . . . . . 8 ⊢ ({𝑛} ∈ 𝒫 No → {𝑛} <<s ∅)
17	14, 15, 16	3syl 19	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → {𝑛} <<s ∅)
18		un0 4343	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
19	18	imaeq2i 6049	. . . . . . . . 9 ⊢ (bday “ ({𝑛} ∪ ∅)) = (bday “ {𝑛})
20		bdayfn 28139	. . . . . . . . . 10 ⊢ bday Fn No
21		fnsnfv 6964	. . . . . . . . . 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 2814	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → (bday “ ({𝑛} ∪ ∅)) = {(bday‘𝑛)})
24		fvex 6898	. . . . . . . . . 10 ⊢ (bday‘𝑛) ∈ V
25	24	sucid 6447	. . . . . . . . 9 ⊢ (bday‘𝑛) ∈ suc (bday‘𝑛)
26		snssi 4742	. . . . . . . . 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 3952	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (bday “ ({𝑛} ∪ ∅)) ⊆ suc (bday‘𝑛))
29		bdayon 28143	. . . . . . . . 9 ⊢ (bday‘𝑛) ∈ On
30	29	onsuci 7850	. . . . . . . 8 ⊢ suc (bday‘𝑛) ∈ On
31		cutbdaybnd 28186	. . . . . . . 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 3942	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → (bday‘(𝑛 +s 1s )) ⊆ suc (bday‘𝑛))
35		bdayon 28143	. . . . . 6 ⊢ (bday‘(𝑛 +s 1s )) ∈ On
36		onsssuc 6455	. . . . . 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 7901	. . . . 5 ⊢ ((bday‘𝑛) ∈ ω → suc (bday‘𝑛) ∈ ω)
40		peano2 7901	. . . . 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 7888	. . . 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 28724	1 ⊢ (𝐴 ∈ ℕ0s → (bday‘𝐴) ∈ ω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  𝒫 cpw 4556  {csn 4584   class class class wbr 5102   “ cima 5653  Oncon0 6362  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  ωcom 7877  Nocsur 28002  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150   0_s c0s 28196   1_s c1s 28197   +_s cadds 28350  ℕ_0scn0s 28703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-1s 28199  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-subs 28413  df-n0s 28705

This theorem is used by:  n0ssoldg  28744  eln0s2  28748  onltn0s  28749  bdayn0sf1o  28761  zsbday  28797  bdayfinbndlem1  28858  z12bdaylem  28875