Theorem bdayn0p1 28383

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 28349	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) = ({𝐴} |s ∅))
2	1	fveq2d 6835	. 2 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = ( bday ‘({𝐴} |s ∅)))
3		n0no 28337	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
4		snelpwi 5386	. . . . 5 ⊢ (𝐴 ∈ No → {𝐴} ∈ 𝒫 No )
5		nulsgts 27790	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 No → {𝐴} <<s ∅)
6	3, 4, 5	3syl 18	. . . 4 ⊢ (𝐴 ∈ ℕ0s → {𝐴} <<s ∅)
7		un0 4325	. . . . . 6 ⊢ ({𝐴} ∪ ∅) = {𝐴}
8	7	imaeq2i 6017	. . . . 5 ⊢ ( bday “ ({𝐴} ∪ ∅)) = ( bday “ {𝐴})
9		bdayfn 27763	. . . . . . . 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 4111	. . . . . . 7 ⊢ {( bday ‘𝐴)} ⊆ (( bday ‘𝐴) ∪ {( bday ‘𝐴)})
13		df-suc 6320	. . . . . . 7 ⊢ suc ( bday ‘𝐴) = (( bday ‘𝐴) ∪ {( bday ‘𝐴)})
14	12, 13	sseqtrri 3966	. . . . . 6 ⊢ {( bday ‘𝐴)} ⊆ suc ( bday ‘𝐴)
15	11, 14	eqsstrdi 3961	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ( bday “ {𝐴}) ⊆ suc ( bday ‘𝐴))
16	8, 15	eqsstrid 3955	. . . 4 ⊢ (𝐴 ∈ ℕ0s → ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴))
17		bdayon 27766	. . . . . 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 27809	. . . . 5 ⊢ (({𝐴} <<s ∅ ∧ suc ( bday ‘𝐴) ∈ On ∧ ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴)) → ( bday ‘({𝐴} |s ∅)) ⊆ suc ( bday ‘𝐴))
21	19, 20	mp3an2 1458	. . . 4 ⊢ (({𝐴} <<s ∅ ∧ ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴)) → ( bday ‘({𝐴} |s ∅)) ⊆ suc ( bday ‘𝐴))
22	6, 16, 21	syl2anc 591	. . 3 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘({𝐴} |s ∅)) ⊆ suc ( bday ‘𝐴))
23		sltssep 27781	. . . . . . . 8 ⊢ ({𝐴} <<s {𝑧} → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦)
24		breq1 5078	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦))
25	24	ralbidv 3164	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝑧}𝐴 <s 𝑦))
26		vex 3437	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
27		breq2 5079	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝐴 <s 𝑦 ↔ 𝐴 <s 𝑧))
28	26, 27	ralsn 4616	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ {𝑧}𝐴 <s 𝑦 ↔ 𝐴 <s 𝑧)
29	25, 28	bitrdi 289	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ 𝐴 <s 𝑧))
30	29	ralsng 4610	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ0s → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ 𝐴 <s 𝑧))
31	30	adantr 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ) → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ 𝐴 <s 𝑧))
32		n0on 28350	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
33		onnolt 28280	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Ons ∧ 𝑧 ∈ No ∧ 𝐴 <s 𝑧) → ( bday ‘𝐴) ∈ ( bday ‘𝑧))
34	32, 33	syl3an1 1170	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ∧ 𝐴 <s 𝑧) → ( bday ‘𝐴) ∈ ( bday ‘𝑧))
35	34	3expia 1128	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ) → (𝐴 <s 𝑧 → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
36	31, 35	sylbid 242	. . . . . . . 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 493	. . . . . 6 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝑧 ∈ No ) → (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
39	38	ralrimiva 3133	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ∀𝑧 ∈ No (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
40		ssint 4897	. . . . . 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 4014	. . . . . . . 8 ⊢ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
42		sseq2 3943	. . . . . . . . 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 699	. . . . . . 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 27766	. . . . . . . . . 10 ⊢ ( bday ‘𝑧) ∈ On
46	17, 45	onsucssi 7785	. . . . . . . . 9 ⊢ (( bday ‘𝐴) ∈ ( bday ‘𝑧) ↔ suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧))
47	46	ralbii 3087	. . . . . . . 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 4568	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
49	48	breq2d 5087	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ({𝐴} <<s {𝑥} ↔ {𝐴} <<s {𝑧}))
50	48	breq1d 5085	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ({𝑥} <<s ∅ ↔ {𝑧} <<s ∅))
51	49, 50	anbi12d 639	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅) ↔ ({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅)))
52	51	ralrab 3637	. . . . . . . 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 279	. . . . . . 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 277	. . . . . 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 277	. . . . 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 236	. . . 4 ⊢ (𝐴 ∈ ℕ0s → suc ( bday ‘𝐴) ⊆ ∩ ( bday “ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)}))
57		cutbday 27798	. . . . 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 3954	. . 3 ⊢ (𝐴 ∈ ℕ0s → suc ( bday ‘𝐴) ⊆ ( bday ‘({𝐴} |s ∅)))
60	22, 59	eqssd 3934	. 2 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘({𝐴} |s ∅)) = suc ( bday ‘𝐴))
61	2, 60	eqtrd 2776	1 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = suc ( bday ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  {crab 3393   ∪ cun 3883   ⊆ wss 3885  ∅c0 4264  𝒫 cpw 4532  {csn 4558  ∩ cint 4880   class class class wbr 5075   “ cima 5624  Oncon0 6314  suc csuc 6316   Fn wfn 6484  ‘cfv 6489  (class class class)co 7360   No csur 27625   <s clts 27626   bday cbday 27627   <<s cslts 27771   |s ccuts 27773   1_s c1s 27820   +_s cadds 27973  On_scons 28265  ℕ_0scn0s 28326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  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 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774  df-0s 27821  df-1s 27822  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-norec 27952  df-norec2 27963  df-adds 27974  df-negs 28035  df-subs 28036  df-ons 28266  df-n0s 28328

This theorem is referenced by:  bdayn0sf1o  28384  bdaypw2n0bndlem  28477  bdaypw2bnd  28479  bdayfinbndlem1  28481  z12bdaylem2  28485