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Theorem lrold 33707
Description: The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 12-Aug-2024.)
Assertion
Ref Expression
lrold (𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴)))

Proof of Theorem lrold
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 leftval 33676 . . . 4 (𝐴 No → ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴})
2 rightval 33677 . . . 4 (𝐴 No → ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥})
31, 2uneq12d 4052 . . 3 (𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ({𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥}))
4 unrab 4192 . . 3 ({𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥}) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ (𝑥 <s 𝐴𝐴 <s 𝑥)}
53, 4eqtrdi 2789 . 2 (𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ (𝑥 <s 𝐴𝐴 <s 𝑥)})
6 oldirr 33702 . . . . . . 7 ¬ 𝐴 ∈ ( O ‘( bday 𝐴))
7 eleq1 2820 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 ∈ ( O ‘( bday 𝐴)) ↔ 𝐴 ∈ ( O ‘( bday 𝐴))))
86, 7mtbiri 330 . . . . . 6 (𝑥 = 𝐴 → ¬ 𝑥 ∈ ( O ‘( bday 𝐴)))
98necon2ai 2963 . . . . 5 (𝑥 ∈ ( O ‘( bday 𝐴)) → 𝑥𝐴)
109adantl 485 . . . 4 ((𝐴 No 𝑥 ∈ ( O ‘( bday 𝐴))) → 𝑥𝐴)
11 bdayelon 33604 . . . . . . 7 ( bday 𝐴) ∈ On
12 oldf 33674 . . . . . . . . 9 O :On⟶𝒫 No
1312ffvelrni 6854 . . . . . . . 8 (( bday 𝐴) ∈ On → ( O ‘( bday 𝐴)) ∈ 𝒫 No )
1413elpwid 4496 . . . . . . 7 (( bday 𝐴) ∈ On → ( O ‘( bday 𝐴)) ⊆ No )
1511, 14ax-mp 5 . . . . . 6 ( O ‘( bday 𝐴)) ⊆ No
1615sseli 3871 . . . . 5 (𝑥 ∈ ( O ‘( bday 𝐴)) → 𝑥 No )
17 slttrine 33587 . . . . . 6 ((𝑥 No 𝐴 No ) → (𝑥𝐴 ↔ (𝑥 <s 𝐴𝐴 <s 𝑥)))
1817ancoms 462 . . . . 5 ((𝐴 No 𝑥 No ) → (𝑥𝐴 ↔ (𝑥 <s 𝐴𝐴 <s 𝑥)))
1916, 18sylan2 596 . . . 4 ((𝐴 No 𝑥 ∈ ( O ‘( bday 𝐴))) → (𝑥𝐴 ↔ (𝑥 <s 𝐴𝐴 <s 𝑥)))
2010, 19mpbid 235 . . 3 ((𝐴 No 𝑥 ∈ ( O ‘( bday 𝐴))) → (𝑥 <s 𝐴𝐴 <s 𝑥))
2120rabeqcda 3394 . 2 (𝐴 No → {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ (𝑥 <s 𝐴𝐴 <s 𝑥)} = ( O ‘( bday 𝐴)))
225, 21eqtrd 2773 1 (𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 846   = wceq 1542  wcel 2113  wne 2934  {crab 3057  cun 3839  wss 3841  𝒫 cpw 4485   class class class wbr 5027  Oncon0 6166  cfv 6333   No csur 33476   <s cslt 33477   bday cbday 33478   O cold 33660   L cleft 33662   R cright 33663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-wrecs 7969  df-recs 8030  df-1o 8124  df-2o 8125  df-no 33479  df-slt 33480  df-bday 33481  df-sslt 33609  df-scut 33611  df-made 33664  df-old 33665  df-left 33667  df-right 33668
This theorem is referenced by:  lruneq  33716  lrrecval2  33726
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