Theorem onsbnd 28578

Description: The surreals of a given birthday are bounded above by that ordinal. (Contributed by Scott Fenton, 22-Feb-2026.)

Assertion

Ref	Expression
onsbnd	⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ≤s 𝐴)

Proof of Theorem onsbnd

Dummy variables 𝑥_𝑅 𝑦_𝐿 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4454	. . 3 ⊢ ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅)
3		leftssold 28168	. . . . . . 7 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
4		bdayon 28049	. . . . . . . 8 ⊢ ( bday ‘𝐴) ∈ On
5		madebdayim 28185	. . . . . . . . 9 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
6	5	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
7		oldss 28167	. . . . . . . 8 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ⊆ ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) ⊆ ( O ‘( bday ‘𝐴)))
8	4, 6, 7	sylancr 599	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( O ‘( bday ‘𝐵)) ⊆ ( O ‘( bday ‘𝐴)))
9	3, 8	sstrid 3942	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐴)))
10		onleft 28557	. . . . . . 7 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
11	10	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
12	9, 11	sseqtrd 3967	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) ⊆ ( L ‘𝐴))
13	12	sselda 3931	. . . 4 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) ∧ 𝑦𝐿 ∈ ( L ‘𝐵)) → 𝑦𝐿 ∈ ( L ‘𝐴))
14		leftlt 28150	. . . 4 ⊢ (𝑦𝐿 ∈ ( L ‘𝐴) → 𝑦𝐿 <s 𝐴)
15	13, 14	syl 18	. . 3 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) ∧ 𝑦𝐿 ∈ ( L ‘𝐵)) → 𝑦𝐿 <s 𝐴)
16	15	ralrimiva 3154	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ∀𝑦𝐿 ∈ ( L ‘𝐵)𝑦𝐿 <s 𝐴)
17		lltr 28159	. . . 4 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
18	17	a1i 11	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) <<s ( R ‘𝐵))
19		leftssno 28170	. . . . 5 ⊢ ( L ‘𝐴) ⊆ No
20		fvex 6894	. . . . . 6 ⊢ ( L ‘𝐴) ∈ V
21	20	elpw 4561	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
22	19, 21	mpbir 234	. . . 4 ⊢ ( L ‘𝐴) ∈ 𝒫 No
23		nulsgts 28073	. . . 4 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
24	22, 23	mp1i 14	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐴) <<s ∅)
25		madeno 28140	. . . . . 6 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → 𝐵 ∈ No )
26	25	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ∈ No )
27		lrcut 28201	. . . . 5 ⊢ (𝐵 ∈ No → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
28	26, 27	syl 18	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
29	28	eqcomd 2766	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 = (( L ‘𝐵) |s ( R ‘𝐵)))
30		oncutleft 28560	. . . 4 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
31	30	adantr 486	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐴 = (( L ‘𝐴) |s ∅))
32	18, 24, 29, 31	lesrecd 28097	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (𝐵 ≤s 𝐴 ↔ (∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅 ∧ ∀𝑦𝐿 ∈ ( L ‘𝐵)𝑦𝐿 <s 𝐴)))
33	2, 16, 32	mpbir2and 726	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ≤s 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   class class class wbr 5103  Oncon0 6359  ‘cfv 6535  (class class class)co 7416   No csur 27908   <s clts 27909   bday cbday 27910   ≤s cles 28012   <<s cslts 28054   |s ccuts 28056   M cmade 28119   O cold 28120   L cleft 28122   R cright 28123  On_scons 28548

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-1o 8462  df-2o 8463  df-no 27911  df-lts 27912  df-bday 27913  df-les 28013  df-slts 28055  df-cuts 28057  df-made 28124  df-old 28125  df-left 28127  df-right 28128  df-ons 28549

This theorem is used by:  onsbnd2  28579  bdayfinbndlem1  28764