Theorem onsbnd 28260

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 4450	. . 3 ⊢ ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅
2	1	a1i 11	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅)
3		leftssold 27859	. . . . . . 7 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
4		bdayelon 27750	. . . . . . . 8 ⊢ ( bday ‘𝐴) ∈ On
5		madebdayim 27868	. . . . . . . . 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 27858	. . . . . . . 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 3944	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐴)))
10		onsleft 28239	. . . . . . 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 3969	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) ⊆ ( L ‘𝐴))
13	12	sselda 3932	. . . 4 ⊢ (((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) ∧ 𝑦𝐿 ∈ ( L ‘𝐵)) → 𝑦𝐿 ∈ ( L ‘𝐴))
14		leftlt 27843	. . . 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 3127	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ∀𝑦𝐿 ∈ ( L ‘𝐵)𝑦𝐿 <s 𝐴)
17		lltropt 27852	. . . 4 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
18	17	a1i 11	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐵) <<s ( R ‘𝐵))
19		leftssno 27861	. . . . 5 ⊢ ( L ‘𝐴) ⊆ No
20		fvex 6846	. . . . . 6 ⊢ ( L ‘𝐴) ∈ V
21	20	elpw 4557	. . . . 5 ⊢ (( L ‘𝐴) ∈ 𝒫 No ↔ ( L ‘𝐴) ⊆ No )
22	19, 21	mpbir 231	. . . 4 ⊢ ( L ‘𝐴) ∈ 𝒫 No
23		nulssgt 27774	. . . 4 ⊢ (( L ‘𝐴) ∈ 𝒫 No → ( L ‘𝐴) <<s ∅)
24	22, 23	mp1i 13	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → ( L ‘𝐴) <<s ∅)
25		madessno 27836	. . . . . . 7 ⊢ ( M ‘( bday ‘𝐴)) ⊆ No
26	25	sseli 3928	. . . . . 6 ⊢ (𝐵 ∈ ( M ‘( bday ‘𝐴)) → 𝐵 ∈ No )
27	26	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ∈ No )
28		lrcut 27884	. . . . 5 ⊢ (𝐵 ∈ No → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
29	27, 28	syl 17	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
30	29	eqcomd 2741	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 = (( L ‘𝐵) |s ( R ‘𝐵)))
31		onscutleft 28242	. . . 4 ⊢ (𝐴 ∈ Ons → 𝐴 = (( L ‘𝐴) |s ∅))
32	31	adantr 480	. . 3 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐴 = (( L ‘𝐴) |s ∅))
33	18, 24, 30, 32	slerecd 27796	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → (𝐵 ≤s 𝐴 ↔ (∀𝑥𝑅 ∈ ∅ 𝐵 <s 𝑥𝑅 ∧ ∀𝑦𝐿 ∈ ( L ‘𝐵)𝑦𝐿 <s 𝐴)))
34	2, 16, 33	mpbir2and 714	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ ( M ‘( bday ‘𝐴))) → 𝐵 ≤s 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050   ⊆ wss 3900  ∅c0 4284  𝒫 cpw 4553   class class class wbr 5097  Oncon0 6316  ‘cfv 6491  (class class class)co 7358   No csur 27609   <s cslt 27610   bday cbday 27611   ≤s csle 27714   <<s csslt 27755   |s cscut 27757   M cmade 27818   O cold 27819   L cleft 27821   R cright 27822  On_scons 28230

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-2o 8398  df-no 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-ons 28231

This theorem is referenced by:  onsbnd2  28261  bdayfinbndlem1  28444