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Theorem sltonold 28283
Description: The class of ordinals less than any surreal is a subset of that surreal's old set. (Contributed by Scott Fenton, 22-Mar-2025.)
Assertion
Ref Expression
sltonold (𝐴 No → {𝑥 ∈ Ons𝑥 <s 𝐴} ⊆ ( O ‘( bday 𝐴)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem sltonold
Dummy variable 𝑥𝑂 is distinct from all other variables.
StepHypRef Expression
1 bdayelon 27821 . . . . . . 7 ( bday 𝑥) ∈ On
21onordi 6495 . . . . . 6 Ord ( bday 𝑥)
3 bdayelon 27821 . . . . . . 7 ( bday 𝐴) ∈ On
43onordi 6495 . . . . . 6 Ord ( bday 𝐴)
5 ordtri2or 6482 . . . . . 6 ((Ord ( bday 𝑥) ∧ Ord ( bday 𝐴)) → (( bday 𝑥) ∈ ( bday 𝐴) ∨ ( bday 𝐴) ⊆ ( bday 𝑥)))
62, 4, 5mp2an 692 . . . . 5 (( bday 𝑥) ∈ ( bday 𝐴) ∨ ( bday 𝐴) ⊆ ( bday 𝑥))
76a1i 11 . . . 4 ((𝐴 No 𝑥 ∈ Ons𝑥 <s 𝐴) → (( bday 𝑥) ∈ ( bday 𝐴) ∨ ( bday 𝐴) ⊆ ( bday 𝑥)))
8 madeun 27922 . . . . . . . . . 10 ( M ‘( bday 𝑥)) = (( O ‘( bday 𝑥)) ∪ ( N ‘( bday 𝑥)))
98eleq2i 2833 . . . . . . . . 9 (𝐴 ∈ ( M ‘( bday 𝑥)) ↔ 𝐴 ∈ (( O ‘( bday 𝑥)) ∪ ( N ‘( bday 𝑥))))
10 elun 4153 . . . . . . . . 9 (𝐴 ∈ (( O ‘( bday 𝑥)) ∪ ( N ‘( bday 𝑥))) ↔ (𝐴 ∈ ( O ‘( bday 𝑥)) ∨ 𝐴 ∈ ( N ‘( bday 𝑥))))
119, 10bitri 275 . . . . . . . 8 (𝐴 ∈ ( M ‘( bday 𝑥)) ↔ (𝐴 ∈ ( O ‘( bday 𝑥)) ∨ 𝐴 ∈ ( N ‘( bday 𝑥))))
12 lrold 27935 . . . . . . . . . . 11 (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday 𝑥))
1312eleq2i 2833 . . . . . . . . . 10 (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ↔ 𝐴 ∈ ( O ‘( bday 𝑥)))
14 elons 28276 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Ons ↔ (𝑥 No ∧ ( R ‘𝑥) = ∅))
1514simprbi 496 . . . . . . . . . . . . . . 15 (𝑥 ∈ Ons → ( R ‘𝑥) = ∅)
1615adantl 481 . . . . . . . . . . . . . 14 ((𝐴 No 𝑥 ∈ Ons) → ( R ‘𝑥) = ∅)
1716uneq2d 4168 . . . . . . . . . . . . 13 ((𝐴 No 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( L ‘𝑥) ∪ ∅))
18 un0 4394 . . . . . . . . . . . . 13 (( L ‘𝑥) ∪ ∅) = ( L ‘𝑥)
1917, 18eqtrdi 2793 . . . . . . . . . . . 12 ((𝐴 No 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( L ‘𝑥))
2019eleq2d 2827 . . . . . . . . . . 11 ((𝐴 No 𝑥 ∈ Ons) → (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ↔ 𝐴 ∈ ( L ‘𝑥)))
21 simpll 767 . . . . . . . . . . . . 13 (((𝐴 No 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝐴 No )
22 onsno 28278 . . . . . . . . . . . . . 14 (𝑥 ∈ Ons𝑥 No )
2322ad2antlr 727 . . . . . . . . . . . . 13 (((𝐴 No 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝑥 No )
24 breq1 5146 . . . . . . . . . . . . . . . 16 (𝑥𝑂 = 𝐴 → (𝑥𝑂 <s 𝑥𝐴 <s 𝑥))
25 leftval 27902 . . . . . . . . . . . . . . . 16 ( L ‘𝑥) = {𝑥𝑂 ∈ ( O ‘( bday 𝑥)) ∣ 𝑥𝑂 <s 𝑥}
2624, 25elrab2 3695 . . . . . . . . . . . . . . 15 (𝐴 ∈ ( L ‘𝑥) ↔ (𝐴 ∈ ( O ‘( bday 𝑥)) ∧ 𝐴 <s 𝑥))
2726simprbi 496 . . . . . . . . . . . . . 14 (𝐴 ∈ ( L ‘𝑥) → 𝐴 <s 𝑥)
2827adantl 481 . . . . . . . . . . . . 13 (((𝐴 No 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝐴 <s 𝑥)
2921, 23, 28sltled 27814 . . . . . . . . . . . 12 (((𝐴 No 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝐴 ≤s 𝑥)
3029ex 412 . . . . . . . . . . 11 ((𝐴 No 𝑥 ∈ Ons) → (𝐴 ∈ ( L ‘𝑥) → 𝐴 ≤s 𝑥))
3120, 30sylbid 240 . . . . . . . . . 10 ((𝐴 No 𝑥 ∈ Ons) → (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) → 𝐴 ≤s 𝑥))
3213, 31biimtrrid 243 . . . . . . . . 9 ((𝐴 No 𝑥 ∈ Ons) → (𝐴 ∈ ( O ‘( bday 𝑥)) → 𝐴 ≤s 𝑥))
33 newbday 27940 . . . . . . . . . . . 12 ((( bday 𝑥) ∈ On ∧ 𝐴 No ) → (𝐴 ∈ ( N ‘( bday 𝑥)) ↔ ( bday 𝐴) = ( bday 𝑥)))
341, 33mpan 690 . . . . . . . . . . 11 (𝐴 No → (𝐴 ∈ ( N ‘( bday 𝑥)) ↔ ( bday 𝐴) = ( bday 𝑥)))
3534adantr 480 . . . . . . . . . 10 ((𝐴 No 𝑥 ∈ Ons) → (𝐴 ∈ ( N ‘( bday 𝑥)) ↔ ( bday 𝐴) = ( bday 𝑥)))
36 leftssold 27917 . . . . . . . . . . . . 13 ( L ‘𝐴) ⊆ ( O ‘( bday 𝐴))
37 fveq2 6906 . . . . . . . . . . . . . . 15 (( bday 𝐴) = ( bday 𝑥) → ( O ‘( bday 𝐴)) = ( O ‘( bday 𝑥)))
3837adantl 481 . . . . . . . . . . . . . 14 (((𝐴 No 𝑥 ∈ Ons) ∧ ( bday 𝐴) = ( bday 𝑥)) → ( O ‘( bday 𝐴)) = ( O ‘( bday 𝑥)))
3915uneq2d 4168 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Ons → (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( L ‘𝑥) ∪ ∅))
4039, 12, 183eqtr3g 2800 . . . . . . . . . . . . . . 15 (𝑥 ∈ Ons → ( O ‘( bday 𝑥)) = ( L ‘𝑥))
4140ad2antlr 727 . . . . . . . . . . . . . 14 (((𝐴 No 𝑥 ∈ Ons) ∧ ( bday 𝐴) = ( bday 𝑥)) → ( O ‘( bday 𝑥)) = ( L ‘𝑥))
4238, 41eqtr2d 2778 . . . . . . . . . . . . 13 (((𝐴 No 𝑥 ∈ Ons) ∧ ( bday 𝐴) = ( bday 𝑥)) → ( L ‘𝑥) = ( O ‘( bday 𝐴)))
4336, 42sseqtrrid 4027 . . . . . . . . . . . 12 (((𝐴 No 𝑥 ∈ Ons) ∧ ( bday 𝐴) = ( bday 𝑥)) → ( L ‘𝐴) ⊆ ( L ‘𝑥))
44 slelss 27946 . . . . . . . . . . . . . 14 ((𝐴 No 𝑥 No ∧ ( bday 𝐴) = ( bday 𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
4522, 44syl3an2 1165 . . . . . . . . . . . . 13 ((𝐴 No 𝑥 ∈ Ons ∧ ( bday 𝐴) = ( bday 𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
46453expa 1119 . . . . . . . . . . . 12 (((𝐴 No 𝑥 ∈ Ons) ∧ ( bday 𝐴) = ( bday 𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
4743, 46mpbird 257 . . . . . . . . . . 11 (((𝐴 No 𝑥 ∈ Ons) ∧ ( bday 𝐴) = ( bday 𝑥)) → 𝐴 ≤s 𝑥)
4847ex 412 . . . . . . . . . 10 ((𝐴 No 𝑥 ∈ Ons) → (( bday 𝐴) = ( bday 𝑥) → 𝐴 ≤s 𝑥))
4935, 48sylbid 240 . . . . . . . . 9 ((𝐴 No 𝑥 ∈ Ons) → (𝐴 ∈ ( N ‘( bday 𝑥)) → 𝐴 ≤s 𝑥))
5032, 49jaod 860 . . . . . . . 8 ((𝐴 No 𝑥 ∈ Ons) → ((𝐴 ∈ ( O ‘( bday 𝑥)) ∨ 𝐴 ∈ ( N ‘( bday 𝑥))) → 𝐴 ≤s 𝑥))
5111, 50biimtrid 242 . . . . . . 7 ((𝐴 No 𝑥 ∈ Ons) → (𝐴 ∈ ( M ‘( bday 𝑥)) → 𝐴 ≤s 𝑥))
52 madebday 27938 . . . . . . . . 9 ((( bday 𝑥) ∈ On ∧ 𝐴 No ) → (𝐴 ∈ ( M ‘( bday 𝑥)) ↔ ( bday 𝐴) ⊆ ( bday 𝑥)))
531, 52mpan 690 . . . . . . . 8 (𝐴 No → (𝐴 ∈ ( M ‘( bday 𝑥)) ↔ ( bday 𝐴) ⊆ ( bday 𝑥)))
5453adantr 480 . . . . . . 7 ((𝐴 No 𝑥 ∈ Ons) → (𝐴 ∈ ( M ‘( bday 𝑥)) ↔ ( bday 𝐴) ⊆ ( bday 𝑥)))
55 slenlt 27797 . . . . . . . 8 ((𝐴 No 𝑥 No ) → (𝐴 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝐴))
5622, 55sylan2 593 . . . . . . 7 ((𝐴 No 𝑥 ∈ Ons) → (𝐴 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝐴))
5751, 54, 563imtr3d 293 . . . . . 6 ((𝐴 No 𝑥 ∈ Ons) → (( bday 𝐴) ⊆ ( bday 𝑥) → ¬ 𝑥 <s 𝐴))
5857con2d 134 . . . . 5 ((𝐴 No 𝑥 ∈ Ons) → (𝑥 <s 𝐴 → ¬ ( bday 𝐴) ⊆ ( bday 𝑥)))
59583impia 1118 . . . 4 ((𝐴 No 𝑥 ∈ Ons𝑥 <s 𝐴) → ¬ ( bday 𝐴) ⊆ ( bday 𝑥))
607, 59olcnd 878 . . 3 ((𝐴 No 𝑥 ∈ Ons𝑥 <s 𝐴) → ( bday 𝑥) ∈ ( bday 𝐴))
61223ad2ant2 1135 . . . 4 ((𝐴 No 𝑥 ∈ Ons𝑥 <s 𝐴) → 𝑥 No )
62 oldbday 27939 . . . 4 ((( bday 𝐴) ∈ On ∧ 𝑥 No ) → (𝑥 ∈ ( O ‘( bday 𝐴)) ↔ ( bday 𝑥) ∈ ( bday 𝐴)))
633, 61, 62sylancr 587 . . 3 ((𝐴 No 𝑥 ∈ Ons𝑥 <s 𝐴) → (𝑥 ∈ ( O ‘( bday 𝐴)) ↔ ( bday 𝑥) ∈ ( bday 𝐴)))
6460, 63mpbird 257 . 2 ((𝐴 No 𝑥 ∈ Ons𝑥 <s 𝐴) → 𝑥 ∈ ( O ‘( bday 𝐴)))
6564rabssdv 4075 1 (𝐴 No → {𝑥 ∈ Ons𝑥 <s 𝐴} ⊆ ( O ‘( bday 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  {crab 3436  cun 3949  wss 3951  c0 4333   class class class wbr 5143  Ord word 6383  Oncon0 6384  cfv 6561   No csur 27684   <s cslt 27685   bday cbday 27686   ≤s csle 27789   M cmade 27881   O cold 27882   N cnew 27883   L cleft 27884   R cright 27885  Onscons 28274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-made 27886  df-old 27887  df-new 27888  df-left 27889  df-right 27890  df-ons 28275
This theorem is referenced by:  sltonex  28284
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