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Theorem noetalem5 33105
 Description: Lemma for noeta 33106. 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 3518 . . 3 (𝐴𝑉𝐴 ∈ V)
21anim2i 616 . 2 ((𝐴 No 𝐴𝑉) → (𝐴 No 𝐴 ∈ V))
3 elex 3518 . . 3 (𝐵𝑊𝐵 ∈ V)
43anim2i 616 . 2 ((𝐵 No 𝐵𝑊) → (𝐵 No 𝐵 ∈ V))
5 id 22 . 2 (∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏 → ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏)
6 simp1l 1191 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝐴 No )
7 simp1r 1192 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝐴 ∈ V)
8 simp2r 1194 . . . 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 33101 . . . 4 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )
126, 7, 8, 11syl3anc 1365 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝑍 No )
13 simplll 771 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝐴 No )
14 simpllr 772 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝐴 ∈ V)
15 simplrr 774 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝐵 ∈ V)
16 simpr 485 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝑎𝐴)
179, 10noetalem2 33102 . . . . . 6 (((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑎𝐴) → 𝑎 <s 𝑍)
1813, 14, 15, 16, 17syl31anc 1367 . . . . 5 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝑎 <s 𝑍)
1918ralrimiva 3187 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ∀𝑎𝐴 𝑎 <s 𝑍)
20193adant3 1126 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∀𝑎𝐴 𝑎 <s 𝑍)
219, 10noetalem3 33103 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∀𝑏𝐵 𝑍 <s 𝑏)
229, 10noetalem4 33104 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))
23223adant3 1126 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))
24 breq2 5067 . . . . . 6 (𝑧 = 𝑍 → (𝑎 <s 𝑧𝑎 <s 𝑍))
2524ralbidv 3202 . . . . 5 (𝑧 = 𝑍 → (∀𝑎𝐴 𝑎 <s 𝑧 ↔ ∀𝑎𝐴 𝑎 <s 𝑍))
26 breq1 5066 . . . . . 6 (𝑧 = 𝑍 → (𝑧 <s 𝑏𝑍 <s 𝑏))
2726ralbidv 3202 . . . . 5 (𝑧 = 𝑍 → (∀𝑏𝐵 𝑧 <s 𝑏 ↔ ∀𝑏𝐵 𝑍 <s 𝑏))
28 fveq2 6667 . . . . . 6 (𝑧 = 𝑍 → ( bday 𝑧) = ( bday 𝑍))
2928sseq1d 4002 . . . . 5 (𝑧 = 𝑍 → (( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵)) ↔ ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵))))
3025, 27, 293anbi123d 1429 . . . 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 3627 . . 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 1366 . 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 1154 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 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  {cab 2804  ∀wral 3143  ∃wrex 3144  Vcvv 3500   ∖ cdif 3937   ∪ cun 3938   ⊆ wss 3940  ifcif 4470  {csn 4564  ⟨cop 4570  ∪ cuni 4837   class class class wbr 5063   ↦ cmpt 5143   × cxp 5552  dom cdm 5554   ↾ cres 5556   “ cima 5557  suc csuc 6191  ℩cio 6310  ‘cfv 6352  ℩crio 7105  1oc1o 8086  2oc2o 8087   No csur 33031
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