Theorem onsbnd 28694

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 4457	. . 3 ⊢ ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅)
3		leftssold 28284	. . . . . . 7 ⊢ (L‘𝐵) ⊆ (O‘(bday‘𝐵))
4		bdayon 28165	. . . . . . . 8 ⊢ (bday‘𝐴) ∈ On
5		madebdayim 28301	. . . . . . . . 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 28283	. . . . . . . 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 3924	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → (L‘𝐵) ⊆ (O‘(bday‘𝐴)))
10		onleft 28673	. . . . . . 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 3949	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → (L‘𝐵) ⊆ (L‘𝐴))
13	12	sselda 3913	. . . 4 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) ∧ 𝑦𝐿 ∈ (L‘𝐵)) → 𝑦𝐿 ∈ (L‘𝐴))
14		leftlt 28266	. . . 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 3158	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → ∀𝑦𝐿 ∈ (L‘𝐵)𝑦𝐿 <s 𝐴)
17		lltr 28275	. . . 4 ⊢ (L‘𝐵) <<s (R‘𝐵)
18	17	a1i 11	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → (L‘𝐵) <<s (R‘𝐵))
19		leftssno 28286	. . . . 5 ⊢ (L‘𝐴) ⊆ No
20		fvex 6918	. . . . . 6 ⊢ (L‘𝐴) ∈ V
21	20	elpw 4564	. . . . 5 ⊢ ((L‘𝐴) ∈ 𝒫 No ↔ (L‘𝐴) ⊆ No)
22	19, 21	mpbir 234	. . . 4 ⊢ (L‘𝐴) ∈ 𝒫 No
23		nulsgts 28189	. . . 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 28256	. . . . . 6 ⊢ (𝐵 ∈ (M‘(bday‘𝐴)) → 𝐵 ∈ No)
26	25	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → 𝐵 ∈ No)
27		lrcut 28317	. . . . 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 2769	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → 𝐵 = ((L‘𝐵) |s (R‘𝐵)))
30		oncutleft 28676	. . . 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 28213	. 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 2147  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560   class class class wbr 5106  Oncon0 6377  ‘cfv 6555  (class class class)co 7440  Nocsur 28024   <s clts 28025  bdaycbday 28026   ≤s cles 28128   <<s cslts 28170   |s ccuts 28172  Mcmade 28235  Ocold 28236  Lcleft 28238  Rcright 28239  On_scons 28664

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-ons 28665

This theorem is used by:  onsbnd2  28695  bdayfinbndlem1  28880