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Theorem sltonold 28298
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 27836 . . . . . . 7 ( bday 𝑥) ∈ On
21onordi 6497 . . . . . 6 Ord ( bday 𝑥)
3 bdayelon 27836 . . . . . . 7 ( bday 𝐴) ∈ On
43onordi 6497 . . . . . 6 Ord ( bday 𝐴)
5 ordtri2or 6484 . . . . . 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 27937 . . . . . . . . . 10 ( M ‘( bday 𝑥)) = (( O ‘( bday 𝑥)) ∪ ( N ‘( bday 𝑥)))
98eleq2i 2831 . . . . . . . . 9 (𝐴 ∈ ( M ‘( bday 𝑥)) ↔ 𝐴 ∈ (( O ‘( bday 𝑥)) ∪ ( N ‘( bday 𝑥))))
10 elun 4163 . . . . . . . . 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 27950 . . . . . . . . . . 11 (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday 𝑥))
1312eleq2i 2831 . . . . . . . . . 10 (𝐴 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ↔ 𝐴 ∈ ( O ‘( bday 𝑥)))
14 elons 28291 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Ons ↔ (𝑥 No ∧ ( R ‘𝑥) = ∅))
1514simprbi 496 . . . . . . . . . . . . . . 15 (𝑥 ∈ Ons → ( R ‘𝑥) = ∅)
1615adantl 481 . . . . . . . . . . . . . 14 ((𝐴 No 𝑥 ∈ Ons) → ( R ‘𝑥) = ∅)
1716uneq2d 4178 . . . . . . . . . . . . 13 ((𝐴 No 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( L ‘𝑥) ∪ ∅))
18 un0 4400 . . . . . . . . . . . . 13 (( L ‘𝑥) ∪ ∅) = ( L ‘𝑥)
1917, 18eqtrdi 2791 . . . . . . . . . . . 12 ((𝐴 No 𝑥 ∈ Ons) → (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( L ‘𝑥))
2019eleq2d 2825 . . . . . . . . . . 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 28293 . . . . . . . . . . . . . 14 (𝑥 ∈ Ons𝑥 No )
2322ad2antlr 727 . . . . . . . . . . . . 13 (((𝐴 No 𝑥 ∈ Ons) ∧ 𝐴 ∈ ( L ‘𝑥)) → 𝑥 No )
24 breq1 5151 . . . . . . . . . . . . . . . 16 (𝑥𝑂 = 𝐴 → (𝑥𝑂 <s 𝑥𝐴 <s 𝑥))
25 leftval 27917 . . . . . . . . . . . . . . . 16 ( L ‘𝑥) = {𝑥𝑂 ∈ ( O ‘( bday 𝑥)) ∣ 𝑥𝑂 <s 𝑥}
2624, 25elrab2 3698 . . . . . . . . . . . . . . 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 27829 . . . . . . . . . . . 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 27955 . . . . . . . . . . . 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 27932 . . . . . . . . . . . . 13 ( L ‘𝐴) ⊆ ( O ‘( bday 𝐴))
37 fveq2 6907 . . . . . . . . . . . . . . 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 4178 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Ons → (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( L ‘𝑥) ∪ ∅))
4039, 12, 183eqtr3g 2798 . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . 13 (((𝐴 No 𝑥 ∈ Ons) ∧ ( bday 𝐴) = ( bday 𝑥)) → ( L ‘𝑥) = ( O ‘( bday 𝐴)))
4336, 42sseqtrrid 4049 . . . . . . . . . . . 12 (((𝐴 No 𝑥 ∈ Ons) ∧ ( bday 𝐴) = ( bday 𝑥)) → ( L ‘𝐴) ⊆ ( L ‘𝑥))
44 slelss 27961 . . . . . . . . . . . . . 14 ((𝐴 No 𝑥 No ∧ ( bday 𝐴) = ( bday 𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
4522, 44syl3an2 1163 . . . . . . . . . . . . 13 ((𝐴 No 𝑥 ∈ Ons ∧ ( bday 𝐴) = ( bday 𝑥)) → (𝐴 ≤s 𝑥 ↔ ( L ‘𝐴) ⊆ ( L ‘𝑥)))
46453expa 1117 . . . . . . . . . . . 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 859 . . . . . . . 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 27953 . . . . . . . . 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 27812 . . . . . . . 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 1116 . . . 4 ((𝐴 No 𝑥 ∈ Ons𝑥 <s 𝐴) → ¬ ( bday 𝐴) ⊆ ( bday 𝑥))
607, 59olcnd 877 . . 3 ((𝐴 No 𝑥 ∈ Ons𝑥 <s 𝐴) → ( bday 𝑥) ∈ ( bday 𝐴))
61223ad2ant2 1133 . . . 4 ((𝐴 No 𝑥 ∈ Ons𝑥 <s 𝐴) → 𝑥 No )
62 oldbday 27954 . . . 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 4085 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 847  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  cun 3961  wss 3963  c0 4339   class class class wbr 5148  Ord word 6385  Oncon0 6386  cfv 6563   No csur 27699   <s cslt 27700   bday cbday 27701   ≤s csle 27804   M cmade 27896   O cold 27897   N cnew 27898   L cleft 27899   R cright 27900  Onscons 28289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902  df-new 27903  df-left 27904  df-right 27905  df-ons 28290
This theorem is referenced by:  sltonex  28299
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