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Theorem nosupbnd1lem6 27759
Description: Lemma for nosupbnd1 27760. 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 1199 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝐴 No )
2 simp3 1138 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑈𝐴)
31, 2sseldd 3983 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑈 No )
4 nofv 27703 . . . . 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 1108 . . . 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 218 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ¬ (¬ (𝑈‘dom 𝑆) = ∅ ∧ ¬ (𝑈‘dom 𝑆) = 1o ∧ ¬ (𝑈‘dom 𝑆) = 2o))
8 simpl1 1191 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦)
9 simpl2 1192 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝐴 No 𝐴 ∈ V))
10 simpl3 1193 . . . . . 6 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → 𝑈𝐴)
11 simpr 484 . . . . . 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 27757 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅)
148, 9, 10, 11, 13syl112anc 1375 . . . . 5 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ ∅)
1514neneqd 2944 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = ∅)
1612nosupbnd1lem5 27758 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 1o)
178, 9, 10, 11, 16syl112anc 1375 . . . . 5 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ 1o)
1817neneqd 2944 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = 1o)
1912nosupbnd1lem3 27756 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ (𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2o)
208, 9, 10, 11, 19syl112anc 1375 . . . . 5 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → (𝑈‘dom 𝑆) ≠ 2o)
2120neneqd 2944 . . . 4 (((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) ∧ (𝑈 ↾ dom 𝑆) = 𝑆) → ¬ (𝑈‘dom 𝑆) = 2o)
2215, 18, 213jca 1128 . . 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 27754 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆))
2512nosupno 27749 . . . . . 6 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
26253ad2ant2 1134 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → 𝑆 No )
27 nodmon 27696 . . . . 5 (𝑆 No → dom 𝑆 ∈ On)
2826, 27syl 17 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → dom 𝑆 ∈ On)
29 noreson 27706 . . . 4 ((𝑈 No ∧ dom 𝑆 ∈ On) → (𝑈 ↾ dom 𝑆) ∈ No )
303, 28, 29syl2anc 584 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑈𝐴) → (𝑈 ↾ dom 𝑆) ∈ No )
31 sltso 27722 . . . 4 <s Or No
32 solin 5618 . . . 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 690 . . 3 (((𝑈 ↾ dom 𝑆) ∈ No 𝑆 No ) → ((𝑈 ↾ dom 𝑆) <s 𝑆 ∨ (𝑈 ↾ dom 𝑆) = 𝑆𝑆 <s (𝑈 ↾ dom 𝑆)))
3430, 26, 33syl2anc 584 . 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 1474 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 395  w3o 1085  w3a 1086   = wceq 1539  wcel 2107  {cab 2713  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cun 3948  wss 3950  c0 4332  ifcif 4524  {csn 4625  cop 4631   class class class wbr 5142  cmpt 5224   Or wor 5590  dom cdm 5684  cres 5686  Oncon0 6383  suc csuc 6385  cio 6511  cfv 6560  crio 7388  1oc1o 8500  2oc2o 8501   No csur 27685   <s cslt 27686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-riota 7389  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689  df-bday 27690
This theorem is referenced by:  nosupbnd1  27760
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