Theorem bdayn0p1 28299

Description: The birthday of 𝐴 +_s 1_s is the successor of the birthday of 𝐴 when 𝐴 is a non-negative surreal integer. (Contributed by Scott Fenton, 7-Nov-2025.)

Assertion

Ref	Expression
bdayn0p1	⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = suc ( bday ‘𝐴))

Proof of Theorem bdayn0p1

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0scut2 28268	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) = ({𝐴} |s ∅))
2	1	fveq2d 6844	. 2 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = ( bday ‘({𝐴} |s ∅)))
3		n0sno 28257	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
4		snelpwi 5398	. . . . 5 ⊢ (𝐴 ∈ No → {𝐴} ∈ 𝒫 No )
5		nulssgt 27745	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 No → {𝐴} <<s ∅)
6	3, 4, 5	3syl 18	. . . 4 ⊢ (𝐴 ∈ ℕ0s → {𝐴} <<s ∅)
7		un0 4353	. . . . . 6 ⊢ ({𝐴} ∪ ∅) = {𝐴}
8	7	imaeq2i 6018	. . . . 5 ⊢ ( bday “ ({𝐴} ∪ ∅)) = ( bday “ {𝐴})
9		bdayfn 27719	. . . . . . . 8 ⊢ bday Fn No
10	9	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → bday Fn No )
11	10, 3	fnimasnd 7322	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ( bday “ {𝐴}) = {( bday ‘𝐴)})
12		ssun2 4138	. . . . . . 7 ⊢ {( bday ‘𝐴)} ⊆ (( bday ‘𝐴) ∪ {( bday ‘𝐴)})
13		df-suc 6326	. . . . . . 7 ⊢ suc ( bday ‘𝐴) = (( bday ‘𝐴) ∪ {( bday ‘𝐴)})
14	12, 13	sseqtrri 3993	. . . . . 6 ⊢ {( bday ‘𝐴)} ⊆ suc ( bday ‘𝐴)
15	11, 14	eqsstrdi 3988	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ( bday “ {𝐴}) ⊆ suc ( bday ‘𝐴))
16	8, 15	eqsstrid 3982	. . . 4 ⊢ (𝐴 ∈ ℕ0s → ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴))
17		bdayelon 27722	. . . . . 6 ⊢ ( bday ‘𝐴) ∈ On
18		onsuc 7767	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ On → suc ( bday ‘𝐴) ∈ On)
19	17, 18	ax-mp 5	. . . . 5 ⊢ suc ( bday ‘𝐴) ∈ On
20		scutbdaybnd 27762	. . . . 5 ⊢ (({𝐴} <<s ∅ ∧ suc ( bday ‘𝐴) ∈ On ∧ ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴)) → ( bday ‘({𝐴} |s ∅)) ⊆ suc ( bday ‘𝐴))
21	19, 20	mp3an2 1451	. . . 4 ⊢ (({𝐴} <<s ∅ ∧ ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴)) → ( bday ‘({𝐴} |s ∅)) ⊆ suc ( bday ‘𝐴))
22	6, 16, 21	syl2anc 584	. . 3 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘({𝐴} |s ∅)) ⊆ suc ( bday ‘𝐴))
23		ssltsep 27737	. . . . . . . 8 ⊢ ({𝐴} <<s {𝑧} → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦)
24		breq1 5105	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦))
25	24	ralbidv 3156	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝑧}𝐴 <s 𝑦))
26		vex 3448	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
27		breq2 5106	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝐴 <s 𝑦 ↔ 𝐴 <s 𝑧))
28	26, 27	ralsn 4641	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ {𝑧}𝐴 <s 𝑦 ↔ 𝐴 <s 𝑧)
29	25, 28	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ 𝐴 <s 𝑧))
30	29	ralsng 4635	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0s → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ 𝐴 <s 𝑧))
31	30	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ 𝐴 <s 𝑧))
32		n0ons 28269	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
33		onnolt 28208	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝑧 ∈ No ∧ 𝐴 <s 𝑧) → ( bday ‘𝐴) ∈ ( bday ‘𝑧))
34	32, 33	syl3an1 1163	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ∧ 𝐴 <s 𝑧) → ( bday ‘𝐴) ∈ ( bday ‘𝑧))
35	34	3expia 1121	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ) → (𝐴 <s 𝑧 → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
36	31, 35	sylbid 240	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
37	23, 36	syl5 34	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ) → ({𝐴} <<s {𝑧} → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
38	37	adantrd 491	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ) → (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
39	38	ralrimiva 3125	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ∀𝑧 ∈ No (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
40		ssint 4924	. . . . . 6 ⊢ (suc ( bday ‘𝐴) ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑦 ∈ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)})suc ( bday ‘𝐴) ⊆ 𝑦)
41		ssrab2 4039	. . . . . . . 8 ⊢ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
42		sseq2 3970	. . . . . . . . 9 ⊢ (𝑦 = ( bday ‘𝑧) → (suc ( bday ‘𝐴) ⊆ 𝑦 ↔ suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧)))
43	42	ralima 7193	. . . . . . . 8 ⊢ (( bday Fn No ∧ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No ) → (∀𝑦 ∈ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)})suc ( bday ‘𝐴) ⊆ 𝑦 ↔ ∀𝑧 ∈ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧)))
44	9, 41, 43	mp2an 692	. . . . . . 7 ⊢ (∀𝑦 ∈ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)})suc ( bday ‘𝐴) ⊆ 𝑦 ↔ ∀𝑧 ∈ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧))
45		bdayelon 27722	. . . . . . . . . 10 ⊢ ( bday ‘𝑧) ∈ On
46	17, 45	onsucssi 7797	. . . . . . . . 9 ⊢ (( bday ‘𝐴) ∈ ( bday ‘𝑧) ↔ suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧))
47	46	ralbii 3075	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)} ( bday ‘𝐴) ∈ ( bday ‘𝑧) ↔ ∀𝑧 ∈ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧))
48		sneq 4595	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
49	48	breq2d 5114	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ({𝐴} <<s {𝑥} ↔ {𝐴} <<s {𝑧}))
50	48	breq1d 5112	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ({𝑥} <<s ∅ ↔ {𝑧} <<s ∅))
51	49, 50	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅)))
52	51	ralrab 3662	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)} ( bday ‘𝐴) ∈ ( bday ‘𝑧) ↔ ∀𝑧 ∈ No (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
53	47, 52	bitr3i 277	. . . . . . 7 ⊢ (∀𝑧 ∈ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧) ↔ ∀𝑧 ∈ No (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
54	44, 53	bitri 275	. . . . . 6 ⊢ (∀𝑦 ∈ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)})suc ( bday ‘𝐴) ⊆ 𝑦 ↔ ∀𝑧 ∈ No (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
55	40, 54	bitri 275	. . . . 5 ⊢ (suc ( bday ‘𝐴) ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}) ↔ ∀𝑧 ∈ No (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
56	39, 55	sylibr 234	. . . 4 ⊢ (𝐴 ∈ ℕ0s → suc ( bday ‘𝐴) ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}))
57		scutbday 27751	. . . . 5 ⊢ ({𝐴} <<s ∅ → ( bday ‘({𝐴} |s ∅)) = ∩ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}))
58	6, 57	syl 17	. . . 4 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘({𝐴} |s ∅)) = ∩ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}))
59	56, 58	sseqtrrd 3981	. . 3 ⊢ (𝐴 ∈ ℕ0s → suc ( bday ‘𝐴) ⊆ ( bday ‘({𝐴} |s ∅)))
60	22, 59	eqssd 3961	. 2 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘({𝐴} |s ∅)) = suc ( bday ‘𝐴))
61	2, 60	eqtrd 2764	1 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = suc ( bday ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585  ∩ cint 4906   class class class wbr 5102   “ cima 5634  Oncon0 6320  suc csuc 6322   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369   No csur 27585   <s cslt 27586   bday cbday 27587   <<s csslt 27727   |s cscut 27729   1_s c1s 27773   +_s cadds 27907  On_scons 28193  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-ons 28194  df-n0s 28249

This theorem is referenced by:  bdayn0sf1o  28300