Theorem onsbnd 28667

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 4453	. . 3 ⊢ ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅)
3		leftssold 28257	. . . . . . 7 ⊢ (L‘𝐵) ⊆ (O‘(bday‘𝐵))
4		bdayon 28138	. . . . . . . 8 ⊢ (bday‘𝐴) ∈ On
5		madebdayim 28274	. . . . . . . . 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 28256	. . . . . . . 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 3919	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → (L‘𝐵) ⊆ (O‘(bday‘𝐴)))
10		onleft 28646	. . . . . . 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 3944	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → (L‘𝐵) ⊆ (L‘𝐴))
13	12	sselda 3908	. . . 4 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) ∧ 𝑦𝐿 ∈ (L‘𝐵)) → 𝑦𝐿 ∈ (L‘𝐴))
14		leftlt 28239	. . . 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 28248	. . . 4 ⊢ (L‘𝐵) <<s (R‘𝐵)
18	17	a1i 11	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → (L‘𝐵) <<s (R‘𝐵))
19		leftssno 28259	. . . . 5 ⊢ (L‘𝐴) ⊆ No
20		fvex 6898	. . . . . 6 ⊢ (L‘𝐴) ∈ V
21	20	elpw 4560	. . . . 5 ⊢ ((L‘𝐴) ∈ 𝒫 No ↔ (L‘𝐴) ⊆ No)
22	19, 21	mpbir 234	. . . 4 ⊢ (L‘𝐴) ∈ 𝒫 No
23		nulsgts 28162	. . . 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 28229	. . . . . 6 ⊢ (𝐵 ∈ (M‘(bday‘𝐴)) → 𝐵 ∈ No)
26	25	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ (M‘(bday‘𝐴))) → 𝐵 ∈ No)
27		lrcut 28290	. . . . 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 28649	. . . 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 28186	. 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 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556   class class class wbr 5102  Oncon0 6362  ‘cfv 6538  (class class class)co 7420  Nocsur 27997   <s clts 27998  bdaycbday 27999   ≤s cles 28101   <<s cslts 28143   |s ccuts 28145  Mcmade 28208  Ocold 28209  Lcleft 28211  Rcright 28212  On_scons 28637

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-ons 28638

This theorem is used by:  onsbnd2  28668  bdayfinbndlem1  28853