Theorem n0sbday 28285

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 6840	. . 3 ⊢ (𝑚 = 0s → ( bday ‘𝑚) = ( bday ‘ 0s ))
2	1	eleq1d 2813	. 2 ⊢ (𝑚 = 0s → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘ 0s ) ∈ ω))
3		fveq2 6840	. . 3 ⊢ (𝑚 = 𝑛 → ( bday ‘𝑚) = ( bday ‘𝑛))
4	3	eleq1d 2813	. 2 ⊢ (𝑚 = 𝑛 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝑛) ∈ ω))
5		fveq2 6840	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ( bday ‘𝑚) = ( bday ‘(𝑛 +s 1s )))
6	5	eleq1d 2813	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘(𝑛 +s 1s )) ∈ ω))
7		fveq2 6840	. . 3 ⊢ (𝑚 = 𝐴 → ( bday ‘𝑚) = ( bday ‘𝐴))
8	7	eleq1d 2813	. 2 ⊢ (𝑚 = 𝐴 → (( bday ‘𝑚) ∈ ω ↔ ( bday ‘𝐴) ∈ ω))
9		bday0s 27778	. . 3 ⊢ ( bday ‘ 0s ) = ∅
10		peano1 7845	. . 3 ⊢ ∅ ∈ ω
11	9, 10	eqeltri 2824	. 2 ⊢ ( bday ‘ 0s ) ∈ ω
12		n0scut2 28268	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) = ({𝑛} |s ∅))
13	12	fveq2d 6844	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) = ( bday ‘({𝑛} |s ∅)))
14		n0sno 28257	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 𝑛 ∈ No )
15		snelpwi 5398	. . . . . . . 8 ⊢ (𝑛 ∈ No → {𝑛} ∈ 𝒫 No )
16		nulssgt 27745	. . . . . . . 8 ⊢ ({𝑛} ∈ 𝒫 No → {𝑛} <<s ∅)
17	14, 15, 16	3syl 18	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → {𝑛} <<s ∅)
18		un0 4353	. . . . . . . . . 10 ⊢ ({𝑛} ∪ ∅) = {𝑛}
19	18	imaeq2i 6018	. . . . . . . . 9 ⊢ ( bday “ ({𝑛} ∪ ∅)) = ( bday “ {𝑛})
20		bdayfn 27719	. . . . . . . . . 10 ⊢ bday Fn No
21		fnsnfv 6922	. . . . . . . . . 10 ⊢ (( bday Fn No ∧ 𝑛 ∈ No ) → {( bday ‘𝑛)} = ( bday “ {𝑛}))
22	20, 14, 21	sylancr 587	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → {( bday ‘𝑛)} = ( bday “ {𝑛}))
23	19, 22	eqtr4id 2783	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) = {( bday ‘𝑛)})
24		fvex 6853	. . . . . . . . . 10 ⊢ ( bday ‘𝑛) ∈ V
25	24	sucid 6404	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ suc ( bday ‘𝑛)
26		snssi 4768	. . . . . . . . 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 3988	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛))
29		bdayelon 27722	. . . . . . . . 9 ⊢ ( bday ‘𝑛) ∈ On
30	29	onsuci 7794	. . . . . . . 8 ⊢ suc ( bday ‘𝑛) ∈ On
31		scutbdaybnd 27762	. . . . . . . 8 ⊢ (({𝑛} <<s ∅ ∧ suc ( bday ‘𝑛) ∈ On ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
32	30, 31	mp3an2 1451	. . . . . . 7 ⊢ (({𝑛} <<s ∅ ∧ ( bday “ ({𝑛} ∪ ∅)) ⊆ suc ( bday ‘𝑛)) → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
33	17, 28, 32	syl2anc 584	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘({𝑛} |s ∅)) ⊆ suc ( bday ‘𝑛))
34	13, 33	eqsstrd 3978	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛))
35		bdayelon 27722	. . . . . 6 ⊢ ( bday ‘(𝑛 +s 1s )) ∈ On
36		onsssuc 6412	. . . . . 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 692	. . . . 5 ⊢ (( bday ‘(𝑛 +s 1s )) ⊆ suc ( bday ‘𝑛) ↔ ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
38	34, 37	sylib 218	. . . 4 ⊢ (𝑛 ∈ ℕ0s → ( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛))
39		peano2 7846	. . . . 5 ⊢ (( bday ‘𝑛) ∈ ω → suc ( bday ‘𝑛) ∈ ω)
40		peano2 7846	. . . . 5 ⊢ (suc ( bday ‘𝑛) ∈ ω → suc suc ( bday ‘𝑛) ∈ ω)
41	39, 40	syl 17	. . . 4 ⊢ (( bday ‘𝑛) ∈ ω → suc suc ( bday ‘𝑛) ∈ ω)
42		elnn 7833	. . . 4 ⊢ ((( bday ‘(𝑛 +s 1s )) ∈ suc suc ( bday ‘𝑛) ∧ suc suc ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
43	38, 41, 42	syl2an 596	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ ( bday ‘𝑛) ∈ ω) → ( bday ‘(𝑛 +s 1s )) ∈ ω)
44	43	ex 412	. 2 ⊢ (𝑛 ∈ ℕ0s → (( bday ‘𝑛) ∈ ω → ( bday ‘(𝑛 +s 1s )) ∈ ω))
45	2, 4, 6, 8, 11, 44	n0sind 28266	1 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585   class class class wbr 5102   “ cima 5634  Oncon0 6320  suc csuc 6322   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369  ωcom 7822   No csur 27585   bday cbday 27587   <<s csslt 27727   |s cscut 27729   0_s c0s 27772   1_s c1s 27773   +_s cadds 27907  ℕ_0scnn0s 28247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-nadd 8607  df-no 27588  df-slt 27589  df-bday 27590  df-sle 27691  df-sslt 27728  df-scut 27730  df-0s 27774  df-1s 27775  df-made 27793  df-old 27794  df-left 27796  df-right 27797  df-norec 27886  df-norec2 27897  df-adds 27908  df-negs 27968  df-subs 27969  df-n0s 28249

This theorem is referenced by:  n0ssold  28286  onltn0s  28289  bdayn0sf1o  28300  zsbday  28335