Theorem n0bday 28771

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 6911	. . 3 ⊢ (𝑚 = 0s → (bday‘𝑚) = (bday‘ 0s ))
2	1	eleq1d 2849	. 2 ⊢ (𝑚 = 0s → ((bday‘𝑚) ∈ ω ↔ (bday‘ 0s ) ∈ ω))
3		fveq2 6911	. . 3 ⊢ (𝑚 = 𝑛 → (bday‘𝑚) = (bday‘𝑛))
4	3	eleq1d 2849	. 2 ⊢ (𝑚 = 𝑛 → ((bday‘𝑚) ∈ ω ↔ (bday‘𝑛) ∈ ω))
5		fveq2 6911	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (bday‘𝑚) = (bday‘(𝑛 +s 1s )))
6	5	eleq1d 2849	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → ((bday‘𝑚) ∈ ω ↔ (bday‘(𝑛 +s 1s )) ∈ ω))
7		fveq2 6911	. . 3 ⊢ (𝑚 = 𝐴 → (bday‘𝑚) = (bday‘𝐴))
8	7	eleq1d 2849	. 2 ⊢ (𝑚 = 𝐴 → ((bday‘𝑚) ∈ ω ↔ (bday‘𝐴) ∈ ω))
9		bday0 28230	. . 3 ⊢ (bday‘ 0s ) = ∅
10		peano1 7927	. . 3 ⊢ ∅ ∈ ω
11	9, 10	eqeltri 2860	. 2 ⊢ (bday‘ 0s ) ∈ ω
12		n0cut2 28754	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({𝑛} |s ∅))
13	12	fveq2d 6915	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → (bday‘(𝑛 +s 1s )) = (bday‘({𝑛} |s ∅)))
14		n0no 28742	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 𝑛 ∈ No)
15		snelpwi 5417	. . . . . . . 8 ⊢ (𝑛 ∈ No → {𝑛} ∈ 𝒫 No)
16		nulsgts 28195	. . . . . . . 8 ⊢ ({𝑛} ∈ 𝒫 No → {𝑛} <<s ∅)
17	14, 15, 16	3syl 19	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → {𝑛} <<s ∅)
18		un0 4347	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
19	18	imaeq2i 6063	. . . . . . . . 9 ⊢ (bday “ ({𝑛} ∪ ∅)) = (bday “ {𝑛})
20		bdayfn 28167	. . . . . . . . . 10 ⊢ bday Fn No
21		fnsnfv 6990	. . . . . . . . . 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 2817	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → (bday “ ({𝑛} ∪ ∅)) = {(bday‘𝑛)})
24		fvex 6924	. . . . . . . . . 10 ⊢ (bday‘𝑛) ∈ V
25	24	sucid 6468	. . . . . . . . 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 3957	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (bday “ ({𝑛} ∪ ∅)) ⊆ suc (bday‘𝑛))
29		bdayon 28171	. . . . . . . . 9 ⊢ (bday‘𝑛) ∈ On
30	29	onsuci 7877	. . . . . . . 8 ⊢ suc (bday‘𝑛) ∈ On
31		cutbdaybnd 28214	. . . . . . . 8 ⊢ (({𝑛} <<s ∅ ∧ suc (bday‘𝑛) ∈ On ∧ (bday “ ({𝑛} ∪ ∅)) ⊆ suc (bday‘𝑛)) → (bday‘({𝑛} |s ∅)) ⊆ suc (bday‘𝑛))
32	30, 31	mp3an2 1479	. . . . . . 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 3947	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → (bday‘(𝑛 +s 1s )) ⊆ suc (bday‘𝑛))
35		bdayon 28171	. . . . . 6 ⊢ (bday‘(𝑛 +s 1s )) ∈ On
36		onsssuc 6476	. . . . . 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 7928	. . . . 5 ⊢ ((bday‘𝑛) ∈ ω → suc (bday‘𝑛) ∈ ω)
40		peano2 7928	. . . . 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 7915	. . . 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 28752	1 ⊢ (𝐴 ∈ ℕ0s → (bday‘𝐴) ∈ ω)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560  {csn 4588   class class class wbr 5106   “ cima 5659  Oncon0 6383  suc csuc 6385   Fn wfn 6556  ‘cfv 6561  (class class class)co 7446  ωcom 7904  Nocsur 28030  bdaycbday 28032   <<s cslts 28176   |s ccuts 28178   0_s c0s 28224   1_s c1s 28225   +_s cadds 28378  ℕ_0scn0s 28731

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-1s 28227  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440  df-subs 28441  df-n0s 28733

This theorem is used by:  n0ssoldg  28772  eln0s2  28776  onltn0s  28777  bdayn0sf1o  28789  zsbday  28825  bdayfinbndlem1  28886  z12bdaylem  28903