Theorem bdayn0p1 28755

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		n0cut2 28721	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) = ({𝐴} |s ∅))
2	1	fveq2d 6889	. 2 ⊢ (𝐴 ∈ ℕ0s → (bday‘(𝐴 +s 1s )) = (bday‘({𝐴} |s ∅)))
3		n0no 28709	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No)
4		snelpwi 5411	. . . . 5 ⊢ (𝐴 ∈ No → {𝐴} ∈ 𝒫 No)
5		nulsgts 28162	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 No → {𝐴} <<s ∅)
6	3, 4, 5	3syl 19	. . . 4 ⊢ (𝐴 ∈ ℕ0s → {𝐴} <<s ∅)
7		un0 4343	. . . . . 6 ⊢ ({𝐴} ∪ ∅) = {𝐴}
8	7	imaeq2i 6049	. . . . 5 ⊢ (bday “ ({𝐴} ∪ ∅)) = (bday “ {𝐴})
9		bdayfn 28134	. . . . . . . 8 ⊢ bday Fn No
10	9	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → bday Fn No)
11	10, 3	fnimasnd 7373	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → (bday “ {𝐴}) = {(bday‘𝐴)})
12		ssun2 4177	. . . . . . 7 ⊢ {(bday‘𝐴)} ⊆ ((bday‘𝐴) ∪ {(bday‘𝐴)})
13		df-suc 6368	. . . . . . 7 ⊢ suc (bday‘𝐴) = ((bday‘𝐴) ∪ {(bday‘𝐴)})
14	12, 13	sseqtrri 3957	. . . . . 6 ⊢ {(bday‘𝐴)} ⊆ suc (bday‘𝐴)
15	11, 14	eqsstrdi 3952	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → (bday “ {𝐴}) ⊆ suc (bday‘𝐴))
16	8, 15	eqsstrid 3946	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (bday “ ({𝐴} ∪ ∅)) ⊆ suc (bday‘𝐴))
17		bdayon 28138	. . . . . 6 ⊢ (bday‘𝐴) ∈ On
18		onsuc 7824	. . . . . 6 ⊢ ((bday‘𝐴) ∈ On → suc (bday‘𝐴) ∈ On)
19	17, 18	ax-mp 5	. . . . 5 ⊢ suc (bday‘𝐴) ∈ On
20		cutbdaybnd 28181	. . . . 5 ⊢ (({𝐴} <<s ∅ ∧ suc (bday‘𝐴) ∈ On ∧ (bday “ ({𝐴} ∪ ∅)) ⊆ suc (bday‘𝐴)) → (bday‘({𝐴} |s ∅)) ⊆ suc (bday‘𝐴))
21	19, 20	mp3an2 1478	. . . 4 ⊢ (({𝐴} <<s ∅ ∧ (bday “ ({𝐴} ∪ ∅)) ⊆ suc (bday‘𝐴)) → (bday‘({𝐴} |s ∅)) ⊆ suc (bday‘𝐴))
22	6, 16, 21	syl2anc 596	. . 3 ⊢ (𝐴 ∈ ℕ0s → (bday‘({𝐴} |s ∅)) ⊆ suc (bday‘𝐴))
23		sltssep 28153	. . . . . . . 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 3185	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝑧}𝐴 <s 𝑦))
26		vex 3454	. . . . . . . . . . . . 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 290	. . . . . . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ 𝐴 <s 𝑧))
32		n0on 28722	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
33		onnolt 28652	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝑧 ∈ No ∧ 𝐴 <s 𝑧) → (bday‘𝐴) ∈ (bday‘𝑧))
34	32, 33	syl3an1 1181	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ∧ 𝐴 <s 𝑧) → (bday‘𝐴) ∈ (bday‘𝑧))
35	34	3expia 1139	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No) → (𝐴 <s 𝑧 → (bday‘𝐴) ∈ (bday‘𝑧)))
36	31, 35	sylbid 243	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 → (bday‘𝐴) ∈ (bday‘𝑧)))
37	23, 36	syl5 35	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No) → ({𝐴} <<s {𝑧} → (bday‘𝐴) ∈ (bday‘𝑧)))
38	37	adantrd 497	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No) → (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → (bday‘𝐴) ∈ (bday‘𝑧)))
39	38	ralrimiva 3154	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ∀𝑧 ∈ No (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → (bday‘𝐴) ∈ (bday‘𝑧)))
40		ssint 4923	. . . . . 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 4005	. . . . . . . 8 ⊢ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
42		sseq2 3934	. . . . . . . . 9 ⊢ (𝑦 = (bday‘𝑧) → (suc (bday‘𝐴) ⊆ 𝑦 ↔ suc (bday‘𝐴) ⊆ (bday‘𝑧)))
43	42	ralima 7243	. . . . . . . 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 705	. . . . . . 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		bdayon 28138	. . . . . . . . . 10 ⊢ (bday‘𝑧) ∈ On
46	17, 45	onsucssi 7852	. . . . . . . . 9 ⊢ ((bday‘𝐴) ∈ (bday‘𝑧) ↔ suc (bday‘𝐴) ⊆ (bday‘𝑧))
47	46	ralbii 3108	. . . . . . . 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 4593	. . . . . . . . . . 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 644	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅)))
52	51	ralrab 3651	. . . . . . . 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 280	. . . . . . 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 278	. . . . . 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 278	. . . . 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 237	. . . 4 ⊢ (𝐴 ∈ ℕ0s → suc (bday‘𝐴) ⊆ ∩ (bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}))
57		cutbday 28170	. . . . 5 ⊢ ({𝐴} <<s ∅ → (bday‘({𝐴} |s ∅)) = ∩ (bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}))
58	6, 57	syl 18	. . . 4 ⊢ (𝐴 ∈ ℕ0s → (bday‘({𝐴} |s ∅)) = ∩ (bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}))
59	56, 58	sseqtrrd 3945	. . 3 ⊢ (𝐴 ∈ ℕ0s → suc (bday‘𝐴) ⊆ (bday‘({𝐴} |s ∅)))
60	22, 59	eqssd 3925	. 2 ⊢ (𝐴 ∈ ℕ0s → (bday‘({𝐴} |s ∅)) = suc (bday‘𝐴))
61	2, 60	eqtrd 2795	1 ⊢ (𝐴 ∈ ℕ0s → (bday‘(𝐴 +s 1s )) = suc (bday‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  𝒫 cpw 4556  {csn 4583  ∩ cint 4906   class class class wbr 5102   “ cima 5653  Oncon0 6362  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Nocsur 27997   <s clts 27998  bdaycbday 27999   <<s cslts 28143   |s ccuts 28145   1_s c1s 28192   +_s cadds 28345  On_scons 28637  ℕ_0scn0s 28698

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 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  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 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-ons 28638  df-n0s 28700

This theorem is used by:  bdayn0sf1o  28756  bdaypw2n0bndlem  28849  bdaypw2bnd  28851  bdayfinbndlem1  28853  z12bdaylem2  28857