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Theorem fimax2gtrilemstep 6801
Description: Lemma for fimax2gtri 6802. The induction step. (Contributed by Jim Kingdon, 5-Sep-2022.)
Hypotheses
Ref Expression
fimax2gtri.po (𝜑𝑅 Po 𝐴)
fimax2gtri.tri (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
fimax2gtri.fin (𝜑𝐴 ∈ Fin)
fimax2gtri.n0 (𝜑𝐴 ≠ ∅)
fimax2gtri.ufin (𝜑𝑈 ∈ Fin)
fimax2gtri.uss (𝜑𝑈𝐴)
fimax2gtri.za (𝜑𝑍𝐴)
fimax2gtri.va (𝜑𝑉𝐴)
fimax2gtri.vu (𝜑 → ¬ 𝑉𝑈)
fimax2gtri.zb (𝜑 → ∀𝑦𝑈 ¬ 𝑍𝑅𝑦)
Assertion
Ref Expression
fimax2gtrilemstep (𝜑 → ∃𝑥𝐴𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦)
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝐴,𝑦   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑥,𝑍,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem fimax2gtrilemstep
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fimax2gtri.va . . 3 (𝜑𝑉𝐴)
2 fimax2gtri.za . . 3 (𝜑𝑍𝐴)
3 fimax2gtri.po . . . 4 (𝜑𝑅 Po 𝐴)
4 fimax2gtri.tri . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
53, 4, 2, 1tridc 6800 . . 3 (𝜑DECID 𝑍𝑅𝑉)
61, 2, 5ifcldcd 3511 . 2 (𝜑 → if(𝑍𝑅𝑉, 𝑉, 𝑍) ∈ 𝐴)
7 simplr 520 . . . . . . . 8 ((((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → 𝑍𝑅𝑉)
8 simpr 109 . . . . . . . . . . . 12 ((𝜑𝑍𝑅𝑉) → 𝑍𝑅𝑉)
98iftrued 3485 . . . . . . . . . . 11 ((𝜑𝑍𝑅𝑉) → if(𝑍𝑅𝑉, 𝑉, 𝑍) = 𝑉)
109breq1d 3946 . . . . . . . . . 10 ((𝜑𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤𝑉𝑅𝑤))
1110biimpa 294 . . . . . . . . 9 (((𝜑𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → 𝑉𝑅𝑤)
1211adantllr 473 . . . . . . . 8 ((((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → 𝑉𝑅𝑤)
133ad2antrr 480 . . . . . . . . . 10 (((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) → 𝑅 Po 𝐴)
142ad2antrr 480 . . . . . . . . . 10 (((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) → 𝑍𝐴)
151ad2antrr 480 . . . . . . . . . 10 (((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) → 𝑉𝐴)
16 fimax2gtri.uss . . . . . . . . . . . 12 (𝜑𝑈𝐴)
1716ad2antrr 480 . . . . . . . . . . 11 (((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) → 𝑈𝐴)
18 simplr 520 . . . . . . . . . . 11 (((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) → 𝑤𝑈)
1917, 18sseldd 3102 . . . . . . . . . 10 (((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) → 𝑤𝐴)
20 potr 4237 . . . . . . . . . 10 ((𝑅 Po 𝐴 ∧ (𝑍𝐴𝑉𝐴𝑤𝐴)) → ((𝑍𝑅𝑉𝑉𝑅𝑤) → 𝑍𝑅𝑤))
2113, 14, 15, 19, 20syl13anc 1219 . . . . . . . . 9 (((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) → ((𝑍𝑅𝑉𝑉𝑅𝑤) → 𝑍𝑅𝑤))
2221adantr 274 . . . . . . . 8 ((((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → ((𝑍𝑅𝑉𝑉𝑅𝑤) → 𝑍𝑅𝑤))
237, 12, 22mp2and 430 . . . . . . 7 ((((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → 𝑍𝑅𝑤)
24 fimax2gtri.zb . . . . . . . . . 10 (𝜑 → ∀𝑦𝑈 ¬ 𝑍𝑅𝑦)
25 breq2 3940 . . . . . . . . . . . 12 (𝑦 = 𝑤 → (𝑍𝑅𝑦𝑍𝑅𝑤))
2625notbid 657 . . . . . . . . . . 11 (𝑦 = 𝑤 → (¬ 𝑍𝑅𝑦 ↔ ¬ 𝑍𝑅𝑤))
2726cbvralv 2657 . . . . . . . . . 10 (∀𝑦𝑈 ¬ 𝑍𝑅𝑦 ↔ ∀𝑤𝑈 ¬ 𝑍𝑅𝑤)
2824, 27sylib 121 . . . . . . . . 9 (𝜑 → ∀𝑤𝑈 ¬ 𝑍𝑅𝑤)
2928r19.21bi 2523 . . . . . . . 8 ((𝜑𝑤𝑈) → ¬ 𝑍𝑅𝑤)
3029ad2antrr 480 . . . . . . 7 ((((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) ∧ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤) → ¬ 𝑍𝑅𝑤)
3123, 30pm2.65da 651 . . . . . 6 (((𝜑𝑤𝑈) ∧ 𝑍𝑅𝑉) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤)
3229adantr 274 . . . . . . 7 (((𝜑𝑤𝑈) ∧ ¬ 𝑍𝑅𝑉) → ¬ 𝑍𝑅𝑤)
33 simpr 109 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → ¬ 𝑍𝑅𝑉)
3433iffalsed 3488 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → if(𝑍𝑅𝑉, 𝑉, 𝑍) = 𝑍)
3534breq1d 3946 . . . . . . . 8 ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤𝑍𝑅𝑤))
3635adantlr 469 . . . . . . 7 (((𝜑𝑤𝑈) ∧ ¬ 𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤𝑍𝑅𝑤))
3732, 36mtbird 663 . . . . . 6 (((𝜑𝑤𝑈) ∧ ¬ 𝑍𝑅𝑉) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤)
38 exmiddc 822 . . . . . . . 8 (DECID 𝑍𝑅𝑉 → (𝑍𝑅𝑉 ∨ ¬ 𝑍𝑅𝑉))
395, 38syl 14 . . . . . . 7 (𝜑 → (𝑍𝑅𝑉 ∨ ¬ 𝑍𝑅𝑉))
4039adantr 274 . . . . . 6 ((𝜑𝑤𝑈) → (𝑍𝑅𝑉 ∨ ¬ 𝑍𝑅𝑉))
4131, 37, 40mpjaodan 788 . . . . 5 ((𝜑𝑤𝑈) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤)
4241ralrimiva 2508 . . . 4 (𝜑 → ∀𝑤𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤)
43 breq2 3940 . . . . . 6 (𝑤 = 𝑦 → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤 ↔ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
4443notbid 657 . . . . 5 (𝑤 = 𝑦 → (¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
4544cbvralv 2657 . . . 4 (∀𝑤𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑤 ↔ ∀𝑦𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
4642, 45sylib 121 . . 3 (𝜑 → ∀𝑦𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
473adantr 274 . . . . . . 7 ((𝜑𝑍𝑅𝑉) → 𝑅 Po 𝐴)
481adantr 274 . . . . . . 7 ((𝜑𝑍𝑅𝑉) → 𝑉𝐴)
49 poirr 4236 . . . . . . 7 ((𝑅 Po 𝐴𝑉𝐴) → ¬ 𝑉𝑅𝑉)
5047, 48, 49syl2anc 409 . . . . . 6 ((𝜑𝑍𝑅𝑉) → ¬ 𝑉𝑅𝑉)
519breq1d 3946 . . . . . 6 ((𝜑𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉𝑉𝑅𝑉))
5250, 51mtbird 663 . . . . 5 ((𝜑𝑍𝑅𝑉) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉)
5334breq1d 3946 . . . . . 6 ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉𝑍𝑅𝑉))
5433, 53mtbird 663 . . . . 5 ((𝜑 ∧ ¬ 𝑍𝑅𝑉) → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉)
5552, 54, 39mpjaodan 788 . . . 4 (𝜑 → ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉)
56 breq2 3940 . . . . . . 7 (𝑦 = 𝑉 → (if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ↔ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉))
5756notbid 657 . . . . . 6 (𝑦 = 𝑉 → (¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉))
5857ralsng 3570 . . . . 5 (𝑉𝐴 → (∀𝑦 ∈ {𝑉} ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉))
591, 58syl 14 . . . 4 (𝜑 → (∀𝑦 ∈ {𝑉} ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑉))
6055, 59mpbird 166 . . 3 (𝜑 → ∀𝑦 ∈ {𝑉} ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
61 ralun 3262 . . 3 ((∀𝑦𝑈 ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦 ∧ ∀𝑦 ∈ {𝑉} ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦) → ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
6246, 60, 61syl2anc 409 . 2 (𝜑 → ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦)
63 breq1 3939 . . . . 5 (𝑥 = if(𝑍𝑅𝑉, 𝑉, 𝑍) → (𝑥𝑅𝑦 ↔ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
6463notbid 657 . . . 4 (𝑥 = if(𝑍𝑅𝑉, 𝑉, 𝑍) → (¬ 𝑥𝑅𝑦 ↔ ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
6564ralbidv 2438 . . 3 (𝑥 = if(𝑍𝑅𝑉, 𝑉, 𝑍) → (∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦))
6665rspcev 2792 . 2 ((if(𝑍𝑅𝑉, 𝑉, 𝑍) ∈ 𝐴 ∧ ∀𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ if(𝑍𝑅𝑉, 𝑉, 𝑍)𝑅𝑦) → ∃𝑥𝐴𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦)
676, 62, 66syl2anc 409 1 (𝜑 → ∃𝑥𝐴𝑦 ∈ (𝑈 ∪ {𝑉}) ¬ 𝑥𝑅𝑦)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 820  w3o 962   = wceq 1332  wcel 1481  wne 2309  wral 2417  wrex 2418  cun 3073  wss 3075  c0 3367  ifcif 3478  {csn 3531   class class class wbr 3936   Po wpo 4223  Fincfn 6641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-if 3479  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-po 4225
This theorem is referenced by:  fimax2gtri  6802
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