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Theorem nosupbnd1lem6 33273
Description: Lemma for nosupbnd1 33274. Establish a hard upper bound when there is no maximum. (Contributed by Scott Fenton, 6-Dec-2021.)
Hypothesis
Ref Expression
nosupbnd1.1 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
nosupbnd1lem6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
Distinct variable groups:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦   𝑢,𝑈,𝑣   𝑥,𝑢,𝑦,𝑣
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑥,𝑦,𝑔)

Proof of Theorem nosupbnd1lem6
StepHypRef Expression
1 simp2l 1196 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝐴 No )
2 simp3 1135 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑈𝐴)
31, 2sseldd 3954 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑈 No )
4 nofv 33224 . . . . 5 (𝑈 No → ((𝑈‘dom 𝑆) = ∅ ∨ (𝑈‘dom 𝑆) = 1o ∨ (𝑈‘dom 𝑆) = 2o))
53, 4syl 17 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ((𝑈‘dom 𝑆) = ∅ ∨ (𝑈‘dom 𝑆) = 1o ∨ (𝑈‘dom 𝑆) = 2o))
6 3oran 1106 . . . 4 (((𝑈‘dom 𝑆) = ∅ ∨ (𝑈‘dom 𝑆) = 1o ∨ (𝑈‘dom 𝑆) = 2o) ↔ ¬ (¬ (𝑈‘dom 𝑆) = ∅ ∧ ¬ (𝑈‘dom 𝑆) = 1o ∧ ¬ (𝑈‘dom 𝑆) = 2o))
75, 6sylib 221 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ¬ (¬ (𝑈‘dom 𝑆) = ∅ ∧ ¬ (𝑈‘dom 𝑆) = 1o ∧ ¬ (𝑈‘dom 𝑆) = 2o))
8 simpl1 1188 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
9 simpl2 1189 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝐴 No 𝐴 ∈ V))
10 simpl3 1190 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → 𝑈𝐴)
11 simpr 488 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈 ↾ dom 𝑆) = 𝑆)
12 nosupbnd1.1 . . . . . . 7 𝑆 = if(∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦, ((𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐴𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
1312nosupbnd1lem4 33271 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅)
148, 9, 10, 11, 13syl112anc 1371 . . . . 5 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ ∅)
1514neneqd 3019 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = ∅)
1612nosupbnd1lem5 33272 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 1o)
178, 9, 10, 11, 16syl112anc 1371 . . . . 5 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ 1o)
1817neneqd 3019 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = 1o)
1912nosupbnd1lem3 33270 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2o)
208, 9, 10, 11, 19syl112anc 1371 . . . . 5 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ 2o)
2120neneqd 3019 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = 2o)
2215, 18, 213jca 1125 . . 3 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (¬ (𝑈‘dom 𝑆) = ∅ ∧ ¬ (𝑈‘dom 𝑆) = 1o ∧ ¬ (𝑈‘dom 𝑆) = 2o))
237, 22mtand 815 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ¬ (𝑈 ↾ dom 𝑆) = 𝑆)
2412nosupbnd1lem1 33268 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
2512nosupno 33263 . . . . . 6 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
26253ad2ant2 1131 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑆 No )
27 nodmon 33217 . . . . 5 (𝑆 No → dom 𝑆 ∈ On)
2826, 27syl 17 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ∈ On)
29 noreson 33227 . . . 4 ((𝑈 No ∧ dom 𝑆 ∈ On) → (𝑈 ↾ dom 𝑆) ∈ No )
303, 28, 29syl2anc 587 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) ∈ No )
31 sltso 33241 . . . 4 <s Or No
32 solin 5485 . . . 4 (( <s Or No ∧ ((𝑈 ↾ dom 𝑆) ∈ No 𝑆 No )) → ((𝑈 ↾ dom 𝑆) <s 𝑆 ∨ (𝑈 ↾ dom 𝑆) = 𝑆𝑆 <s (𝑈 ↾ dom 𝑆)))
3331, 32mpan 689 . . 3 (((𝑈 ↾ dom 𝑆) ∈ No 𝑆 No ) → ((𝑈 ↾ dom 𝑆) <s 𝑆 ∨ (𝑈 ↾ dom 𝑆) = 𝑆𝑆 <s (𝑈 ↾ dom 𝑆)))
3430, 26, 33syl2anc 587 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ((𝑈 ↾ dom 𝑆) <s 𝑆 ∨ (𝑈 ↾ dom 𝑆) = 𝑆𝑆 <s (𝑈 ↾ dom 𝑆)))
3523, 24, 34ecase23d 1470 1 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3o 1083  w3a 1084   = wceq 1538  wcel 2115  {cab 2802  wne 3014  wral 3133  wrex 3134  Vcvv 3480  cun 3917  wss 3919  c0 4276  ifcif 4450  {csn 4550  cop 4556   class class class wbr 5052  cmpt 5132   Or wor 5460  dom cdm 5542  cres 5544  Oncon0 6178  suc csuc 6180  cio 6300  cfv 6343  crio 7106  1oc1o 8091  2oc2o 8092   No csur 33207   <s cslt 33208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-1o 8098  df-2o 8099  df-no 33210  df-slt 33211  df-bday 33212
This theorem is referenced by:  nosupbnd1  33274
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