Theorem onsbnd 28442

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 4464	. . 3 ⊢ ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅)
3		leftssold 28032	. . . . . . 7 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
4		bdayon 27913	. . . . . . . 8 ⊢ ( bday ‘𝐴) ∈ On
5		madebdayim 28049	. . . . . . . . 9 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
6	5	adantl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
7		oldss 28031	. . . . . . . 8 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ⊆ ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) ⊆ ( O ‘( bday ‘𝐴)))
8	4, 6, 7	sylancr 598	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( O ‘( bday ‘𝐵)) ⊆ ( O ‘( bday ‘𝐴)))
9	3, 8	sstrid 3956	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐴)))
10		onleft 28421	. . . . . . 7 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
11	10	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
12	9, 11	sseqtrd 3981	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) ⊆ ( L ‘𝐴))
13	12	sselda 3945	. . . 4 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) ∧ 𝑦𝐿 ∈ ( L ‘𝐵)) → 𝑦𝐿 ∈ ( L ‘𝐴))
14		leftlt 28014	. . . 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 3163	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ∀𝑦𝐿 ∈ ( L ‘𝐵)𝑦𝐿 <s 𝐴)
17		lltr 28023	. . . 4 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
18	17	a1i 11	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) <<s ( R ‘𝐵))
19		leftssno 28034	. . . . 5 ⊢ ( L ‘𝐴) ⊆ No
20		fvex 6897	. . . . . 6 ⊢ ( L ‘𝐴) ∈ V
21	20	elpw 4571	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
22	19, 21	mpbir 234	. . . 4 ⊢ ( L ‘𝐴) ∈ 𝒫 No
23		nulsgts 27937	. . . 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 28004	. . . . . 6 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → 𝐵 ∈ No )
26	25	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ∈ No )
27		lrcut 28065	. . . . 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 2775	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 = (( L ‘𝐵) |s ( R ‘𝐵)))
30		oncutleft 28424	. . . 4 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
31	30	adantr 485	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐴 = (( L ‘𝐴) |s ∅))
32	18, 24, 29, 31	lesrecd 27961	. 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 725	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ≤s 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567   class class class wbr 5113  Oncon0 6363  ‘cfv 6539  (class class class)co 7413   No csur 27772   <s clts 27773   bday cbday 27774   ≤s cles 27876   <<s cslts 27918   |s ccuts 27920   M cmade 27983   O cold 27984   L cleft 27986   R cright 27987  On_scons 28412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-1o 8455  df-2o 8456  df-no 27775  df-lts 27776  df-bday 27777  df-les 27877  df-slts 27919  df-cuts 27921  df-made 27988  df-old 27989  df-left 27991  df-right 27992  df-ons 28413

This theorem is referenced by:  onsbnd2  28443  bdayfinbndlem1  28628