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Theorem nosupbnd1lem2 33334
 Description: Lemma for nosupbnd1 33339. When there is no maximum, if any member of 𝐴 is a prolongment of 𝑆, then so are all elements of 𝐴 above it. (Contributed by Scott Fenton, 5-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
nosupbnd1lem2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) = 𝑆)
Distinct variable groups:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦   𝑣,𝑊   𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑈(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑊(𝑥,𝑦,𝑢,𝑔)

Proof of Theorem nosupbnd1lem2
StepHypRef Expression
1 simp3rr 1244 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ 𝑊 <s 𝑈)
2 simp2l 1196 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝐴 No )
3 simp3rl 1243 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑊𝐴)
42, 3sseldd 3916 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑊 No )
5 simp3ll 1241 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑈𝐴)
62, 5sseldd 3916 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑈 No )
7 nosupbnd1.1 . . . . . . . 8 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))))
87nosupno 33328 . . . . . . 7 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
983ad2ant2 1131 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑆 No )
10 nodmon 33282 . . . . . 6 (𝑆 No → dom 𝑆 ∈ On)
119, 10syl 17 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → dom 𝑆 ∈ On)
12 sltres 33294 . . . . 5 ((𝑊 No 𝑈 No ∧ dom 𝑆 ∈ On) → ((𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆) → 𝑊 <s 𝑈))
134, 6, 11, 12syl3anc 1368 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ((𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆) → 𝑊 <s 𝑈))
141, 13mtod 201 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ (𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆))
15 simp3lr 1242 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑈 ↾ dom 𝑆) = 𝑆)
1615breq2d 5042 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ((𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆) ↔ (𝑊 ↾ dom 𝑆) <s 𝑆))
1714, 16mtbid 327 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ (𝑊 ↾ dom 𝑆) <s 𝑆)
187nosupbnd1lem1 33333 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑊𝐴) → ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))
193, 18syld3an3 1406 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))
20 noreson 33292 . . . 4 ((𝑊 No ∧ dom 𝑆 ∈ On) → (𝑊 ↾ dom 𝑆) ∈ No )
214, 11, 20syl2anc 587 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) ∈ No )
22 sltso 33306 . . . 4 <s Or No
23 sotrieq2 5467 . . . 4 (( <s Or No ∧ ((𝑊 ↾ dom 𝑆) ∈ No 𝑆 No )) → ((𝑊 ↾ dom 𝑆) = 𝑆 ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))))
2422, 23mpan 689 . . 3 (((𝑊 ↾ dom 𝑆) ∈ No 𝑆 No ) → ((𝑊 ↾ dom 𝑆) = 𝑆 ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))))
2521, 9, 24syl2anc 587 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ((𝑊 ↾ dom 𝑆) = 𝑆 ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))))
2617, 19, 25mpbir2and 712 1 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  ifcif 4425  {csn 4525  ⟨cop 4531   class class class wbr 5030   ↦ cmpt 5110   Or wor 5437  dom cdm 5519   ↾ cres 5521  Oncon0 6159  suc csuc 6161  ℩cio 6281  ‘cfv 6324  ℩crio 7092  2oc2o 8081   No csur 33272
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