Theorem onsbnd 28273

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 4438	. . 3 ⊢ ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅)
3		leftssold 27863	. . . . . . 7 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
4		bdayon 27744	. . . . . . . 8 ⊢ ( bday ‘𝐴) ∈ On
5		madebdayim 27880	. . . . . . . . 9 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
6	5	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( bday ‘𝐵) ⊆ ( bday ‘𝐴))
7		oldss 27862	. . . . . . . 8 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ⊆ ( bday ‘𝐴)) → ( O ‘( bday ‘𝐵)) ⊆ ( O ‘( bday ‘𝐴)))
8	4, 6, 7	sylancr 588	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( O ‘( bday ‘𝐵)) ⊆ ( O ‘( bday ‘𝐴)))
9	3, 8	sstrid 3933	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐴)))
10		onleft 28252	. . . . . . 7 ⊢ (𝐴 ∈ Ons → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
11	10	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
12	9, 11	sseqtrd 3958	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) ⊆ ( L ‘𝐴))
13	12	sselda 3921	. . . 4 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) ∧ 𝑦𝐿 ∈ ( L ‘𝐵)) → 𝑦𝐿 ∈ ( L ‘𝐴))
14		leftlt 27845	. . . 4 ⊢ (𝑦𝐿 ∈ ( L ‘𝐴) → 𝑦𝐿 <s 𝐴)
15	13, 14	syl 17	. . 3 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) ∧ 𝑦𝐿 ∈ ( L ‘𝐵)) → 𝑦𝐿 <s 𝐴)
16	15	ralrimiva 3129	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ∀𝑦𝐿 ∈ ( L ‘𝐵)𝑦𝐿 <s 𝐴)
17		lltr 27854	. . . 4 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
18	17	a1i 11	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) <<s ( R ‘𝐵))
19		leftssno 27865	. . . . 5 ⊢ ( L ‘𝐴) ⊆ No
20		fvex 6853	. . . . . 6 ⊢ ( L ‘𝐴) ∈ V
21	20	elpw 4545	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
22	19, 21	mpbir 231	. . . 4 ⊢ ( L ‘𝐴) ∈ 𝒫 No
23		nulsgts 27768	. . . 4 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
24	22, 23	mp1i 13	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐴) <<s ∅)
25		madeno 27835	. . . . . 6 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → 𝐵 ∈ No )
26	25	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ∈ No )
27		lrcut 27896	. . . . 5 ⊢ (𝐵 ∈ No → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
28	26, 27	syl 17	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
29	28	eqcomd 2742	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 = (( L ‘𝐵) |s ( R ‘𝐵)))
30		oncutleft 28255	. . . 4 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
31	30	adantr 480	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐴 = (( L ‘𝐴) |s ∅))
32	18, 24, 29, 31	lesrecd 27792	. 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 714	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ≤s 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541   class class class wbr 5085  Oncon0 6323  ‘cfv 6498  (class class class)co 7367   No csur 27603   <s clts 27604   bday cbday 27605   ≤s cles 27708   <<s cslts 27749   |s ccuts 27751   M cmade 27814   O cold 27815   L cleft 27817   R cright 27818  On_scons 28243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-ons 28244

This theorem is referenced by:  onsbnd2  28274  bdayfinbndlem1  28459