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Theorem lrrecpo 34098
Description: Now, we establish that 𝑅 is a partial ordering on No . (Contributed by Scott Fenton, 19-Aug-2024.)
Hypothesis
Ref Expression
lrrec.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
Assertion
Ref Expression
lrrecpo 𝑅 Po No
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)

Proof of Theorem lrrecpo
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bdayelon 33971 . . . . . 6 ( bday 𝑎) ∈ On
21onirri 6373 . . . . 5 ¬ ( bday 𝑎) ∈ ( bday 𝑎)
3 lrrec.1 . . . . . . 7 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
43lrrecval2 34097 . . . . . 6 ((𝑎 No 𝑎 No ) → (𝑎𝑅𝑎 ↔ ( bday 𝑎) ∈ ( bday 𝑎)))
54anidms 567 . . . . 5 (𝑎 No → (𝑎𝑅𝑎 ↔ ( bday 𝑎) ∈ ( bday 𝑎)))
62, 5mtbiri 327 . . . 4 (𝑎 No → ¬ 𝑎𝑅𝑎)
76adantl 482 . . 3 ((⊤ ∧ 𝑎 No ) → ¬ 𝑎𝑅𝑎)
8 bdayelon 33971 . . . . . 6 ( bday 𝑐) ∈ On
9 ontr1 6312 . . . . . 6 (( bday 𝑐) ∈ On → ((( bday 𝑎) ∈ ( bday 𝑏) ∧ ( bday 𝑏) ∈ ( bday 𝑐)) → ( bday 𝑎) ∈ ( bday 𝑐)))
108, 9ax-mp 5 . . . . 5 ((( bday 𝑎) ∈ ( bday 𝑏) ∧ ( bday 𝑏) ∈ ( bday 𝑐)) → ( bday 𝑎) ∈ ( bday 𝑐))
113lrrecval2 34097 . . . . . . . 8 ((𝑎 No 𝑏 No ) → (𝑎𝑅𝑏 ↔ ( bday 𝑎) ∈ ( bday 𝑏)))
12113adant3 1131 . . . . . . 7 ((𝑎 No 𝑏 No 𝑐 No ) → (𝑎𝑅𝑏 ↔ ( bday 𝑎) ∈ ( bday 𝑏)))
133lrrecval2 34097 . . . . . . . 8 ((𝑏 No 𝑐 No ) → (𝑏𝑅𝑐 ↔ ( bday 𝑏) ∈ ( bday 𝑐)))
14133adant1 1129 . . . . . . 7 ((𝑎 No 𝑏 No 𝑐 No ) → (𝑏𝑅𝑐 ↔ ( bday 𝑏) ∈ ( bday 𝑐)))
1512, 14anbi12d 631 . . . . . 6 ((𝑎 No 𝑏 No 𝑐 No ) → ((𝑎𝑅𝑏𝑏𝑅𝑐) ↔ (( bday 𝑎) ∈ ( bday 𝑏) ∧ ( bday 𝑏) ∈ ( bday 𝑐))))
163lrrecval2 34097 . . . . . . 7 ((𝑎 No 𝑐 No ) → (𝑎𝑅𝑐 ↔ ( bday 𝑎) ∈ ( bday 𝑐)))
17163adant2 1130 . . . . . 6 ((𝑎 No 𝑏 No 𝑐 No ) → (𝑎𝑅𝑐 ↔ ( bday 𝑎) ∈ ( bday 𝑐)))
1815, 17imbi12d 345 . . . . 5 ((𝑎 No 𝑏 No 𝑐 No ) → (((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐) ↔ ((( bday 𝑎) ∈ ( bday 𝑏) ∧ ( bday 𝑏) ∈ ( bday 𝑐)) → ( bday 𝑎) ∈ ( bday 𝑐))))
1910, 18mpbiri 257 . . . 4 ((𝑎 No 𝑏 No 𝑐 No ) → ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
2019adantl 482 . . 3 ((⊤ ∧ (𝑎 No 𝑏 No 𝑐 No )) → ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
217, 20ispod 5512 . 2 (⊤ → 𝑅 Po No )
2221mptru 1546 1 𝑅 Po No
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wtru 1540  wcel 2106  cun 3885   class class class wbr 5074  {copab 5136   Po wpo 5501  Oncon0 6266  cfv 6433   No csur 33843   bday cbday 33845   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:  noinds  34102  norecfn  34103  norecov  34104  noxpordpo  34107  no2indslem  34111  no3inds  34115
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