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Theorem lrrecpo 27989
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 27836 . . . . . 6 ( bday 𝑎) ∈ On
21onirri 6499 . . . . 5 ¬ ( bday 𝑎) ∈ ( bday 𝑎)
3 lrrec.1 . . . . . . 7 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
43lrrecval2 27988 . . . . . 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 27836 . . . . . 6 ( bday 𝑐) ∈ On
9 ontr1 6432 . . . . . 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 27988 . . . . . . . 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 27988 . . . . . . . 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 632 . . . . . 6 ((𝑎 No 𝑏 No 𝑐 No ) → ((𝑎𝑅𝑏𝑏𝑅𝑐) ↔ (( bday 𝑎) ∈ ( bday 𝑏) ∧ ( bday 𝑏) ∈ ( bday 𝑐))))
163lrrecval2 27988 . . . . . . 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 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 5606 . 2 (⊤ → 𝑅 Po No )
2221mptru 1544 1 𝑅 Po No
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wtru 1538  wcel 2106  cun 3961   class class class wbr 5148  {copab 5210   Po wpo 5595  Oncon0 6386  cfv 6563   No csur 27699   bday cbday 27701   L cleft 27899   R cright 27900
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-sslt 27841  df-scut 27843  df-made 27901  df-old 27902  df-left 27904  df-right 27905
This theorem is referenced by:  noinds  27993  norecfn  27994  norecov  27995  noxpordpo  27998  no2indslem  28002  no3inds  28006
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