Theorem bdayn0p1 28294

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 28263	. . 3 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) = ({𝐴} |s ∅))
2	1	fveq2d 6826	. 2 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = ( bday ‘({𝐴} |s ∅)))
3		n0sno 28252	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
4		snelpwi 5383	. . . . 5 ⊢ (𝐴 ∈ No → {𝐴} ∈ 𝒫 No )
5		nulssgt 27739	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 No → {𝐴} <<s ∅)
6	3, 4, 5	3syl 18	. . . 4 ⊢ (𝐴 ∈ ℕ0s → {𝐴} <<s ∅)
7		un0 4341	. . . . . 6 ⊢ ({𝐴} ∪ ∅) = {𝐴}
8	7	imaeq2i 6006	. . . . 5 ⊢ ( bday “ ({𝐴} ∪ ∅)) = ( bday “ {𝐴})
9		bdayfn 27712	. . . . . . . 8 ⊢ bday Fn No
10	9	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → bday Fn No )
11	10, 3	fnimasnd 7299	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ( bday “ {𝐴}) = {( bday ‘𝐴)})
12		ssun2 4126	. . . . . . 7 ⊢ {( bday ‘𝐴)} ⊆ (( bday ‘𝐴) ∪ {( bday ‘𝐴)})
13		df-suc 6312	. . . . . . 7 ⊢ suc ( bday ‘𝐴) = (( bday ‘𝐴) ∪ {( bday ‘𝐴)})
14	12, 13	sseqtrri 3979	. . . . . 6 ⊢ {( bday ‘𝐴)} ⊆ suc ( bday ‘𝐴)
15	11, 14	eqsstrdi 3974	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ( bday “ {𝐴}) ⊆ suc ( bday ‘𝐴))
16	8, 15	eqsstrid 3968	. . . 4 ⊢ (𝐴 ∈ ℕ0s → ( bday “ ({𝐴} ∪ ∅)) ⊆ suc ( bday ‘𝐴))
17		bdayelon 27715	. . . . . 6 ⊢ ( bday ‘𝐴) ∈ On
18		onsuc 7743	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ On → suc ( bday ‘𝐴) ∈ On)
19	17, 18	ax-mp 5	. . . . 5 ⊢ suc ( bday ‘𝐴) ∈ On
20		scutbdaybnd 27756	. . . . 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 27730	. . . . . . . 8 ⊢ ({𝐴} <<s {𝑧} → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝑧}𝑥 <s 𝑦)
24		breq1 5092	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐴 → (𝑥 <s 𝑦 ↔ 𝐴 <s 𝑦))
25	24	ralbidv 3155	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ {𝑧}𝑥 <s 𝑦 ↔ ∀𝑦 ∈ {𝑧}𝐴 <s 𝑦))
26		vex 3440	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
27		breq2 5093	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝐴 <s 𝑦 ↔ 𝐴 <s 𝑧))
28	26, 27	ralsn 4631	. . . . . . . . . . . 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 4625	. . . . . . . . . 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 28264	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons)
33		onnolt 28203	. . . . . . . . . . 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 3124	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ∀𝑧 ∈ No (({𝐴} <<s {𝑧} ∧ {𝑧} <<s ∅) → ( bday ‘𝐴) ∈ ( bday ‘𝑧)))
40		ssint 4912	. . . . . 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 4027	. . . . . . . 8 ⊢ {𝑥 ∈ No ∣ ({𝐴} <<s {𝑥} ∧ {𝑥} <<s ∅)} ⊆ No
42		sseq2 3956	. . . . . . . . 9 ⊢ (𝑦 = ( bday ‘𝑧) → (suc ( bday ‘𝐴) ⊆ 𝑦 ↔ suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧)))
43	42	ralima 7171	. . . . . . . 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 27715	. . . . . . . . . 10 ⊢ ( bday ‘𝑧) ∈ On
46	17, 45	onsucssi 7771	. . . . . . . . 9 ⊢ (( bday ‘𝐴) ∈ ( bday ‘𝑧) ↔ suc ( bday ‘𝐴) ⊆ ( bday ‘𝑧))
47	46	ralbii 3078	. . . . . . . 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 4583	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
49	48	breq2d 5101	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → ({𝐴} <<s {𝑥} ↔ {𝐴} <<s {𝑧}))
50	48	breq1d 5099	. . . . . . . . . 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 3648	. . . . . . . 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 27745	. . . . 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 3967	. . 3 ⊢ (𝐴 ∈ ℕ0s → suc ( bday ‘𝐴) ⊆ ( bday ‘({𝐴} |s ∅)))
60	22, 59	eqssd 3947	. 2 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘({𝐴} |s ∅)) = suc ( bday ‘𝐴))
61	2, 60	eqtrd 2766	1 ⊢ (𝐴 ∈ ℕ0s → ( bday ‘(𝐴 +s 1s )) = suc ( bday ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   ∪ cun 3895   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  {csn 4573  ∩ cint 4895   class class class wbr 5089   “ cima 5617  Oncon0 6306  suc csuc 6308   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346   No csur 27578   <s cslt 27579   bday cbday 27580   <<s csslt 27720   |s cscut 27722   1_s c1s 27767   +_s cadds 27902  On_scons 28188  ℕ_0scnn0s 28242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-nadd 8581  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-0s 27768  df-1s 27769  df-made 27788  df-old 27789  df-left 27791  df-right 27792  df-norec 27881  df-norec2 27892  df-adds 27903  df-negs 27963  df-subs 27964  df-ons 28189  df-n0s 28244

This theorem is referenced by:  bdayn0sf1o  28295