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Theorem lrrecpo 27256
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 27119 . . . . . 6 ( bday 𝑎) ∈ On
21onirri 6431 . . . . 5 ¬ ( bday 𝑎) ∈ ( bday 𝑎)
3 lrrec.1 . . . . . . 7 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}
43lrrecval2 27255 . . . . . 6 ((𝑎 No 𝑎 No ) → (𝑎𝑅𝑎 ↔ ( bday 𝑎) ∈ ( bday 𝑎)))
54anidms 568 . . . . 5 (𝑎 No → (𝑎𝑅𝑎 ↔ ( bday 𝑎) ∈ ( bday 𝑎)))
62, 5mtbiri 327 . . . 4 (𝑎 No → ¬ 𝑎𝑅𝑎)
76adantl 483 . . 3 ((⊤ ∧ 𝑎 No ) → ¬ 𝑎𝑅𝑎)
8 bdayelon 27119 . . . . . 6 ( bday 𝑐) ∈ On
9 ontr1 6364 . . . . . 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 27255 . . . . . . . 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 27255 . . . . . . . 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 27255 . . . . . . 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 345 . . . . 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 483 . . 3 ((⊤ ∧ (𝑎 No 𝑏 No 𝑐 No )) → ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
217, 20ispod 5555 . 2 (⊤ → 𝑅 Po No )
2221mptru 1549 1 𝑅 Po No
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wtru 1543  wcel 2107  cun 3909   class class class wbr 5106  {copab 5168   Po wpo 5544  Oncon0 6318  cfv 6497   No csur 26991   bday cbday 26993   L cleft 27178   R cright 27179
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-1o 8413  df-2o 8414  df-no 26994  df-slt 26995  df-bday 26996  df-sslt 27124  df-scut 27126  df-made 27180  df-old 27181  df-left 27183  df-right 27184
This theorem is referenced by:  noinds  27260  norecfn  27261  norecov  27262  noxpordpo  27265  no2indslem  27269  no3inds  27273
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