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Theorem lrrecpo 27974
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 27821 . . . . . 6 ( bday 𝑎) ∈ On
21onirri 6497 . . . . 5 ¬ ( bday 𝑎) ∈ ( bday 𝑎)
3 lrrec.1 . . . . . . 7 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
43lrrecval2 27973 . . . . . 6 ((𝑎 No 𝑎 No ) → (𝑎𝑅𝑎 ↔ ( bday 𝑎) ∈ ( bday 𝑎)))
54anidms 566 . . . . 5 (𝑎 No → (𝑎𝑅𝑎 ↔ ( bday 𝑎) ∈ ( bday 𝑎)))
62, 5mtbiri 327 . . . 4 (𝑎 No → ¬ 𝑎𝑅𝑎)
76adantl 481 . . 3 ((⊤ ∧ 𝑎 No ) → ¬ 𝑎𝑅𝑎)
8 bdayelon 27821 . . . . . 6 ( bday 𝑐) ∈ On
9 ontr1 6430 . . . . . 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 27973 . . . . . . . 8 ((𝑎 No 𝑏 No ) → (𝑎𝑅𝑏 ↔ ( bday 𝑎) ∈ ( bday 𝑏)))
12113adant3 1133 . . . . . . 7 ((𝑎 No 𝑏 No 𝑐 No ) → (𝑎𝑅𝑏 ↔ ( bday 𝑎) ∈ ( bday 𝑏)))
133lrrecval2 27973 . . . . . . . 8 ((𝑏 No 𝑐 No ) → (𝑏𝑅𝑐 ↔ ( bday 𝑏) ∈ ( bday 𝑐)))
14133adant1 1131 . . . . . . 7 ((𝑎 No 𝑏 No 𝑐 No ) → (𝑏𝑅𝑐 ↔ ( bday 𝑏) ∈ ( bday 𝑐)))
1512, 14anbi12d 632 . . . . . 6 ((𝑎 No 𝑏 No 𝑐 No ) → ((𝑎𝑅𝑏𝑏𝑅𝑐) ↔ (( bday 𝑎) ∈ ( bday 𝑏) ∧ ( bday 𝑏) ∈ ( bday 𝑐))))
163lrrecval2 27973 . . . . . . 7 ((𝑎 No 𝑐 No ) → (𝑎𝑅𝑐 ↔ ( bday 𝑎) ∈ ( bday 𝑐)))
17163adant2 1132 . . . . . 6 ((𝑎 No 𝑏 No 𝑐 No ) → (𝑎𝑅𝑐 ↔ ( bday 𝑎) ∈ ( bday 𝑐)))
1815, 17imbi12d 344 . . . . 5 ((𝑎 No 𝑏 No 𝑐 No ) → (((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐) ↔ ((( bday 𝑎) ∈ ( bday 𝑏) ∧ ( bday 𝑏) ∈ ( bday 𝑐)) → ( bday 𝑎) ∈ ( bday 𝑐))))
1910, 18mpbiri 258 . . . 4 ((𝑎 No 𝑏 No 𝑐 No ) → ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
2019adantl 481 . . 3 ((⊤ ∧ (𝑎 No 𝑏 No 𝑐 No )) → ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
217, 20ispod 5601 . 2 (⊤ → 𝑅 Po No )
2221mptru 1547 1 𝑅 Po No
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wtru 1541  wcel 2108  cun 3949   class class class wbr 5143  {copab 5205   Po wpo 5590  Oncon0 6384  cfv 6561   No csur 27684   bday cbday 27686   L cleft 27884   R cright 27885
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-sslt 27826  df-scut 27828  df-made 27886  df-old 27887  df-left 27889  df-right 27890
This theorem is referenced by:  noinds  27978  norecfn  27979  norecov  27980  noxpordpo  27983  no2indslem  27987  no3inds  27991
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