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Theorem noetalem5 31992
Description: Lemma for noeta 31993. The full statement of the theorem with hypotheses. (Contributed by Scott Fenton, 7-Dec-2021.)
Hypotheses
Ref Expression
noetalem.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))))
noetalem.2 𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜}))
Assertion
Ref Expression
noetalem5 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∃𝑧 No (∀𝑎𝐴 𝑎 <s 𝑧 ∧ ∀𝑏𝐵 𝑧 <s 𝑏 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑔   𝑢,𝑎,𝐴,𝑣,𝑥,𝑦   𝑧,𝑎,𝐴   𝐵,𝑎,𝑏   𝑔,𝑏,𝑥   𝑧,𝑏,𝐵   𝑢,𝑔,𝑣,𝑥,𝑦   𝑆,𝑎,𝑔   𝑣,𝑢,𝑥,𝑦   𝑍,𝑎,𝑏,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑆(𝑥,𝑦,𝑧,𝑣,𝑢,𝑏)   𝑉(𝑥,𝑦,𝑧,𝑣,𝑢,𝑔,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑧,𝑣,𝑢,𝑔,𝑎,𝑏)   𝑍(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetalem5
StepHypRef Expression
1 elex 3243 . . 3 (𝐴𝑉𝐴 ∈ V)
21anim2i 592 . 2 ((𝐴 No 𝐴𝑉) → (𝐴 No 𝐴 ∈ V))
3 elex 3243 . . 3 (𝐵𝑊𝐵 ∈ V)
43anim2i 592 . 2 ((𝐵 No 𝐵𝑊) → (𝐵 No 𝐵 ∈ V))
5 id 22 . 2 (∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏 → ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏)
6 simp1l 1105 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝐴 No )
7 simp1r 1106 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝐴 ∈ V)
8 simp2r 1108 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝐵 ∈ V)
9 noetalem.1 . . . . 5 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))))
10 noetalem.2 . . . . 5 𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜}))
119, 10noetalem1 31988 . . . 4 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )
126, 7, 8, 11syl3anc 1366 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝑍 No )
13 simplll 813 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝐴 No )
14 simpllr 815 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝐴 ∈ V)
15 simplrr 818 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝐵 ∈ V)
16 simpr 476 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝑎𝐴)
179, 10noetalem2 31989 . . . . . 6 (((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑎𝐴) → 𝑎 <s 𝑍)
1813, 14, 15, 16, 17syl31anc 1369 . . . . 5 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝑎 <s 𝑍)
1918ralrimiva 2995 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ∀𝑎𝐴 𝑎 <s 𝑍)
20193adant3 1101 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∀𝑎𝐴 𝑎 <s 𝑍)
219, 10noetalem3 31990 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∀𝑏𝐵 𝑍 <s 𝑏)
229, 10noetalem4 31991 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))
23223adant3 1101 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))
24 breq2 4689 . . . . . 6 (𝑧 = 𝑍 → (𝑎 <s 𝑧𝑎 <s 𝑍))
2524ralbidv 3015 . . . . 5 (𝑧 = 𝑍 → (∀𝑎𝐴 𝑎 <s 𝑧 ↔ ∀𝑎𝐴 𝑎 <s 𝑍))
26 breq1 4688 . . . . . 6 (𝑧 = 𝑍 → (𝑧 <s 𝑏𝑍 <s 𝑏))
2726ralbidv 3015 . . . . 5 (𝑧 = 𝑍 → (∀𝑏𝐵 𝑧 <s 𝑏 ↔ ∀𝑏𝐵 𝑍 <s 𝑏))
28 fveq2 6229 . . . . . 6 (𝑧 = 𝑍 → ( bday 𝑧) = ( bday 𝑍))
2928sseq1d 3665 . . . . 5 (𝑧 = 𝑍 → (( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵)) ↔ ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵))))
3025, 27, 293anbi123d 1439 . . . 4 (𝑧 = 𝑍 → ((∀𝑎𝐴 𝑎 <s 𝑧 ∧ ∀𝑏𝐵 𝑧 <s 𝑏 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))) ↔ (∀𝑎𝐴 𝑎 <s 𝑍 ∧ ∀𝑏𝐵 𝑍 <s 𝑏 ∧ ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))))
3130rspcev 3340 . . 3 ((𝑍 No ∧ (∀𝑎𝐴 𝑎 <s 𝑍 ∧ ∀𝑏𝐵 𝑍 <s 𝑏 ∧ ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))) → ∃𝑧 No (∀𝑎𝐴 𝑎 <s 𝑧 ∧ ∀𝑏𝐵 𝑧 <s 𝑏 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
3212, 20, 21, 23, 31syl13anc 1368 . 2 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∃𝑧 No (∀𝑎𝐴 𝑎 <s 𝑧 ∧ ∀𝑏𝐵 𝑧 <s 𝑏 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
332, 4, 5, 32syl3an 1408 1 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∃𝑧 No (∀𝑎𝐴 𝑎 <s 𝑧 ∧ ∀𝑏𝐵 𝑧 <s 𝑏 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  wss 3607  ifcif 4119  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  cres 5145  cima 5146  suc csuc 5763  cio 5887  cfv 5926  crio 6650  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918   <s cslt 31919   bday cbday 31920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922  df-bday 31923
This theorem is referenced by:  noeta  31993
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