Theorem bdayn0p1 28369

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 28335	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) = ({𝐴} |s ∅))
2	1	fveq2d 6839	. 2 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = ( bday ‘({𝐴} |s ∅)))
3		n0no 28323	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
4		snelpwi 5393	. . . . 5 ⊢ (𝐴 ∈ No → {𝐴} ∈ 𝒫 No )
5		nulsgts 27776	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 No → {𝐴} <<s ∅)
6	3, 4, 5	3syl 18	. . . 4 ⊢ (𝐴 ∈ ℕ0s → {𝐴} <<s ∅)
7		un0 4347	. . . . . 6 ⊢ ({𝐴} ∪ ∅) = {𝐴}
8	7	imaeq2i 6018	. . . . 5 ⊢ ( bday “ ({𝐴} ∪ ∅)) = ( bday “ {𝐴})
9		bdayfn 27749	. . . . . . . 8 ⊢ bday Fn No
10	9	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → bday Fn No )
11	10, 3	fnimasnd 7313	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ( bday “ {𝐴}) = {( bday ‘𝐴)})
12		ssun2 4132	. . . . . . 7 ⊢ {( bday ‘𝐴)} ⊆ (( bday ‘𝐴) ∪ {( bday ‘𝐴)})
13		df-suc 6324	. . . . . . 7 ⊢ suc ( bday ‘𝐴) = (( bday ‘𝐴) ∪ {( bday ‘𝐴)})
14	12, 13	sseqtrri 3984	. . . . . 6 ⊢ {( bday ‘𝐴)} ⊆ suc ( bday ‘𝐴)
15	11, 14	eqsstrdi 3979	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ( bday “ {𝐴}) ⊆ suc ( bday ‘𝐴))
16	8, 15	eqsstrid 3973	. . . 4 ⊢ (𝐴 ∈ ℕ0s → ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴))
17		bdayon 27752	. . . . . 6 ⊢ ( bday ‘𝐴) ∈ On
18		onsuc 7757	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ On → suc ( bday ‘𝐴) ∈ On)
19	17, 18	ax-mp 5	. . . . 5 ⊢ suc ( bday ‘𝐴) ∈ On
20		cutbdaybnd 27795	. . . . 5 ⊢ (({𝐴} <<s ∅ ∧ suc ( bday ‘𝐴) ∈ On ∧ ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴)) → ( bday ‘({𝐴} |s ∅)) ⊆ suc ( bday ‘𝐴))
21	19, 20	mp3an2 1452	. . . 4 ⊢ (({𝐴} <<s ∅ ∧ ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴)) → ( bday ‘({𝐴} |s ∅)) ⊆ suc ( bday ‘𝐴))
22	6, 16, 21	syl2anc 585	. . 3 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘({𝐴} |s ∅)) ⊆ suc ( bday ‘𝐴))
23		sltssep 27767	. . . . . . . 8 ⊢ ({𝐴} <<s {𝑧} → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦)
24		breq1 5102	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦))
25	24	ralbidv 3160	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝑧}𝐴 <s 𝑦))
26		vex 3445	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
27		breq2 5103	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝐴 <s 𝑦 ↔ 𝐴 <s 𝑧))
28	26, 27	ralsn 4639	. . . . . . . . . . . 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 4633	. . . . . . . . . 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		n0on 28336	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
33		onnolt 28266	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝑧 ∈ No ∧ 𝐴 <s 𝑧) → ( bday ‘𝐴) ∈ ( bday ‘𝑧))
34	32, 33	syl3an1 1164	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ∧ 𝐴 <s 𝑧) → ( bday ‘𝐴) ∈ ( bday ‘𝑧))
35	34	3expia 1122	. . . . . . . . 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 3129	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ∀𝑧 ∈ No (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
40		ssint 4920	. . . . . 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 4033	. . . . . . . 8 ⊢ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
42		sseq2 3961	. . . . . . . . 9 ⊢ (𝑦 = ( bday ‘𝑧) → (suc ( bday ‘𝐴) ⊆ 𝑦 ↔ suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧)))
43	42	ralima 7185	. . . . . . . 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 693	. . . . . . 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 27752	. . . . . . . . . 10 ⊢ ( bday ‘𝑧) ∈ On
46	17, 45	onsucssi 7785	. . . . . . . . 9 ⊢ (( bday ‘𝐴) ∈ ( bday ‘𝑧) ↔ suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧))
47	46	ralbii 3083	. . . . . . . 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 4591	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
49	48	breq2d 5111	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ({𝐴} <<s {𝑥} ↔ {𝐴} <<s {𝑧}))
50	48	breq1d 5109	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ({𝑥} <<s ∅ ↔ {𝑧} <<s ∅))
51	49, 50	anbi12d 633	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅)))
52	51	ralrab 3653	. . . . . . . 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		cutbday 27784	. . . . 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 3972	. . 3 ⊢ (𝐴 ∈ ℕ0s → suc ( bday ‘𝐴) ⊆ ( bday ‘({𝐴} |s ∅)))
60	22, 59	eqssd 3952	. 2 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘({𝐴} |s ∅)) = suc ( bday ‘𝐴))
61	2, 60	eqtrd 2772	1 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = suc ( bday ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400   ∪ cun 3900   ⊆ wss 3902  ∅c0 4286  𝒫 cpw 4555  {csn 4581  ∩ cint 4903   class class class wbr 5099   “ cima 5628  Oncon0 6318  suc csuc 6320   Fn wfn 6488  ‘cfv 6493  (class class class)co 7360   No csur 27611   <s clts 27612   bday cbday 27613   <<s cslts 27757   |s ccuts 27759   1_s c1s 27806   +_s cadds 27959  On_scons 28251  ℕ_0scn0s 28312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-nadd 8596  df-no 27614  df-lts 27615  df-bday 27616  df-les 27717  df-slts 27758  df-cuts 27760  df-0s 27807  df-1s 27808  df-made 27827  df-old 27828  df-left 27830  df-right 27831  df-norec 27938  df-norec2 27949  df-adds 27960  df-negs 28021  df-subs 28022  df-ons 28252  df-n0s 28314

This theorem is referenced by:  bdayn0sf1o  28370  bdaypw2n0bndlem  28463  bdaypw2bnd  28465  bdayfinbndlem1  28467  z12bdaylem2  28471