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Theorem noinfbnd1lem2 27088
Description: Lemma for noinfbnd1 27093. When there is no minimum, if any member of 𝐵 is a prolongment of 𝑇, then so are all elements below it. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypothesis
Ref Expression
noinfbnd1.1 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
noinfbnd1lem2 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑊 ↾ dom 𝑇) = 𝑇)
Distinct variable groups:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑈   𝑥,𝑢,𝑦   𝑔,𝑉   𝑥,𝑣,𝑦   𝑣,𝑊
Allowed substitution hints:   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑥,𝑦,𝑢,𝑔)   𝑉(𝑥,𝑦,𝑣,𝑢)   𝑊(𝑥,𝑦,𝑢,𝑔)

Proof of Theorem noinfbnd1lem2
StepHypRef Expression
1 simp3rl 1247 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑊𝐵)
2 noinfbnd1.1 . . . 4 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
32noinfbnd1lem1 27087 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ 𝑊𝐵) → ¬ (𝑊 ↾ dom 𝑇) <s 𝑇)
41, 3syld3an3 1410 . 2 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ¬ (𝑊 ↾ dom 𝑇) <s 𝑇)
5 simp3rr 1248 . . . 4 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ¬ 𝑈 <s 𝑊)
6 simp2l 1200 . . . . . 6 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝐵 No )
7 simp3ll 1245 . . . . . 6 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑈𝐵)
86, 7sseldd 3946 . . . . 5 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑈 No )
96, 1sseldd 3946 . . . . 5 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑊 No )
102noinfno 27082 . . . . . . 7 ((𝐵 No 𝐵𝑉) → 𝑇 No )
11103ad2ant2 1135 . . . . . 6 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → 𝑇 No )
12 nodmon 27014 . . . . . 6 (𝑇 No → dom 𝑇 ∈ On)
1311, 12syl 17 . . . . 5 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → dom 𝑇 ∈ On)
14 sltres 27026 . . . . 5 ((𝑈 No 𝑊 No ∧ dom 𝑇 ∈ On) → ((𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇) → 𝑈 <s 𝑊))
158, 9, 13, 14syl3anc 1372 . . . 4 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ((𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇) → 𝑈 <s 𝑊))
165, 15mtod 197 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ¬ (𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇))
17 simp3lr 1246 . . . 4 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑈 ↾ dom 𝑇) = 𝑇)
1817breq1d 5116 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ((𝑈 ↾ dom 𝑇) <s (𝑊 ↾ dom 𝑇) ↔ 𝑇 <s (𝑊 ↾ dom 𝑇)))
1916, 18mtbid 324 . 2 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))
20 noreson 27024 . . . 4 ((𝑊 No ∧ dom 𝑇 ∈ On) → (𝑊 ↾ dom 𝑇) ∈ No )
219, 13, 20syl2anc 585 . . 3 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑊 ↾ dom 𝑇) ∈ No )
22 sltso 27040 . . . 4 <s Or No
23 sotrieq2 5576 . . . 4 (( <s Or No ∧ ((𝑊 ↾ dom 𝑇) ∈ No 𝑇 No )) → ((𝑊 ↾ dom 𝑇) = 𝑇 ↔ (¬ (𝑊 ↾ dom 𝑇) <s 𝑇 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))))
2422, 23mpan 689 . . 3 (((𝑊 ↾ dom 𝑇) ∈ No 𝑇 No ) → ((𝑊 ↾ dom 𝑇) = 𝑇 ↔ (¬ (𝑊 ↾ dom 𝑇) <s 𝑇 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))))
2521, 11, 24syl2anc 585 . 2 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → ((𝑊 ↾ dom 𝑇) = 𝑇 ↔ (¬ (𝑊 ↾ dom 𝑇) <s 𝑇 ∧ ¬ 𝑇 <s (𝑊 ↾ dom 𝑇))))
264, 19, 25mpbir2and 712 1 ((¬ ∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 No 𝐵𝑉) ∧ ((𝑈𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑊 ↾ dom 𝑇) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  cun 3909  wss 3911  ifcif 4487  {csn 4587  cop 4593   class class class wbr 5106  cmpt 5189   Or wor 5545  dom cdm 5634  cres 5636  Oncon0 6318  suc csuc 6320  cio 6447  cfv 6497  crio 7313  1oc1o 8406   No csur 27004   <s cslt 27005
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 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  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-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-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009
This theorem is referenced by:  noinfbnd1lem3  27089  noinfbnd1lem4  27090  noinfbnd1lem5  27091
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