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Theorem nosupbnd1lem2 32182
Description: Lemma for nosupbnd1 32187. 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 1314 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ 𝑊 <s 𝑈)
2 simp2l 1242 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝐴 No )
3 simp3rl 1313 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑊𝐴)
42, 3sseldd 3745 . . . . 5 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑊 No )
5 simp3ll 1311 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑈𝐴)
62, 5sseldd 3745 . . . . 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 32176 . . . . . . 7 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
983ad2ant2 1129 . . . . . 6 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → 𝑆 No )
10 nodmon 32130 . . . . . 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 32142 . . . . 5 ((𝑊 No 𝑈 No ∧ dom 𝑆 ∈ On) → ((𝑊 ↾ dom 𝑆) <s (𝑈 ↾ dom 𝑆) → 𝑊 <s 𝑈))
134, 6, 11, 12syl3anc 1477 . . . 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 189 . . 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 1312 . . . 4 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑈 ↾ dom 𝑆) = 𝑆)
1615breq2d 4816 . . 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 313 . 2 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → ¬ (𝑊 ↾ dom 𝑆) <s 𝑆)
187nosupbnd1lem1 32181 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ 𝑊𝐴) → ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))
193, 18syld3an3 1516 . 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 32140 . . . 4 ((𝑊 No ∧ dom 𝑆 ∈ On) → (𝑊 ↾ dom 𝑆) ∈ No )
214, 11, 20syl2anc 696 . . 3 ((¬ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 No 𝐴 ∈ V) ∧ ((𝑈𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) ∈ No )
22 sltso 32154 . . . 4 <s Or No
23 sotrieq2 5215 . . . 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 708 . . 3 (((𝑊 ↾ dom 𝑆) ∈ No 𝑆 No ) → ((𝑊 ↾ dom 𝑆) = 𝑆 ↔ (¬ (𝑊 ↾ dom 𝑆) <s 𝑆 ∧ ¬ 𝑆 <s (𝑊 ↾ dom 𝑆))))
2521, 9, 24syl2anc 696 . 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 995 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wrex 3051  Vcvv 3340  cun 3713  wss 3715  ifcif 4230  {csn 4321  cop 4327   class class class wbr 4804  cmpt 4881   Or wor 5186  dom cdm 5266  cres 5268  Oncon0 5884  suc csuc 5886  cio 6010  cfv 6049  crio 6774  2𝑜c2o 7724   No csur 32120   <s cslt 32121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-1o 7730  df-2o 7731  df-no 32123  df-slt 32124  df-bday 32125
This theorem is referenced by:  nosupbnd1lem3  32183  nosupbnd1lem4  32184  nosupbnd1lem5  32185
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