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Theorem lrold 34077
Description: The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.)
Assertion
Ref Expression
lrold (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴))

Proof of Theorem lrold
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 leftval 34047 . . . . 5 ( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴}
2 rightval 34048 . . . . 5 ( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥}
31, 2uneq12i 4095 . . . 4 (( L ‘𝐴) ∪ ( R ‘𝐴)) = ({𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥})
4 unrab 4239 . . . 4 ({𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝑥 <s 𝐴} ∪ {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑥}) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ (𝑥 <s 𝐴𝐴 <s 𝑥)}
53, 4eqtri 2766 . . 3 (( L ‘𝐴) ∪ ( R ‘𝐴)) = {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ (𝑥 <s 𝐴𝐴 <s 𝑥)}
6 oldirr 34072 . . . . . . . 8 ¬ 𝐴 ∈ ( O ‘( bday 𝐴))
7 eleq1 2826 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 ∈ ( O ‘( bday 𝐴)) ↔ 𝐴 ∈ ( O ‘( bday 𝐴))))
86, 7mtbiri 327 . . . . . . 7 (𝑥 = 𝐴 → ¬ 𝑥 ∈ ( O ‘( bday 𝐴)))
98necon2ai 2973 . . . . . 6 (𝑥 ∈ ( O ‘( bday 𝐴)) → 𝑥𝐴)
109adantl 482 . . . . 5 ((𝐴 No 𝑥 ∈ ( O ‘( bday 𝐴))) → 𝑥𝐴)
11 oldssno 34045 . . . . . . 7 ( O ‘( bday 𝐴)) ⊆ No
1211sseli 3917 . . . . . 6 (𝑥 ∈ ( O ‘( bday 𝐴)) → 𝑥 No )
13 slttrine 33954 . . . . . . 7 ((𝑥 No 𝐴 No ) → (𝑥𝐴 ↔ (𝑥 <s 𝐴𝐴 <s 𝑥)))
1413ancoms 459 . . . . . 6 ((𝐴 No 𝑥 No ) → (𝑥𝐴 ↔ (𝑥 <s 𝐴𝐴 <s 𝑥)))
1512, 14sylan2 593 . . . . 5 ((𝐴 No 𝑥 ∈ ( O ‘( bday 𝐴))) → (𝑥𝐴 ↔ (𝑥 <s 𝐴𝐴 <s 𝑥)))
1610, 15mpbid 231 . . . 4 ((𝐴 No 𝑥 ∈ ( O ‘( bday 𝐴))) → (𝑥 <s 𝐴𝐴 <s 𝑥))
1716rabeqcda 3429 . . 3 (𝐴 No → {𝑥 ∈ ( O ‘( bday 𝐴)) ∣ (𝑥 <s 𝐴𝐴 <s 𝑥)} = ( O ‘( bday 𝐴)))
185, 17eqtrid 2790 . 2 (𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴)))
19 un0 4324 . . 3 (∅ ∪ ∅) = ∅
20 leftf 34049 . . . . . . 7 L : No ⟶𝒫 No
2120fdmi 6612 . . . . . 6 dom L = No
2221eleq2i 2830 . . . . 5 (𝐴 ∈ dom L ↔ 𝐴 No )
23 ndmfv 6804 . . . . 5 𝐴 ∈ dom L → ( L ‘𝐴) = ∅)
2422, 23sylnbir 331 . . . 4 𝐴 No → ( L ‘𝐴) = ∅)
25 rightf 34050 . . . . . . 7 R : No ⟶𝒫 No
2625fdmi 6612 . . . . . 6 dom R = No
2726eleq2i 2830 . . . . 5 (𝐴 ∈ dom R ↔ 𝐴 No )
28 ndmfv 6804 . . . . 5 𝐴 ∈ dom R → ( R ‘𝐴) = ∅)
2927, 28sylnbir 331 . . . 4 𝐴 No → ( R ‘𝐴) = ∅)
3024, 29uneq12d 4098 . . 3 𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (∅ ∪ ∅))
31 bdaydm 33969 . . . . . . 7 dom bday = No
3231eleq2i 2830 . . . . . 6 (𝐴 ∈ dom bday 𝐴 No )
33 ndmfv 6804 . . . . . 6 𝐴 ∈ dom bday → ( bday 𝐴) = ∅)
3432, 33sylnbir 331 . . . . 5 𝐴 No → ( bday 𝐴) = ∅)
3534fveq2d 6778 . . . 4 𝐴 No → ( O ‘( bday 𝐴)) = ( O ‘∅))
36 old0 34043 . . . 4 ( O ‘∅) = ∅
3735, 36eqtrdi 2794 . . 3 𝐴 No → ( O ‘( bday 𝐴)) = ∅)
3819, 30, 373eqtr4a 2804 . 2 𝐴 No → (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴)))
3918, 38pm2.61i 182 1 (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  wo 844   = wceq 1539  wcel 2106  wne 2943  {crab 3068  cun 3885  c0 4256  𝒫 cpw 4533   class class class wbr 5074  dom cdm 5589  cfv 6433   No csur 33843   <s cslt 33844   bday cbday 33845   O cold 34027   L cleft 34029   R cright 34030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978  df-made 34031  df-old 34032  df-left 34034  df-right 34035
This theorem is referenced by:  lruneq  34086  lrrecval2  34097
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