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Theorem noetalem5 32380
Description: Lemma for noeta 32381. 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 3400 . . 3 (𝐴𝑉𝐴 ∈ V)
21anim2i 611 . 2 ((𝐴 No 𝐴𝑉) → (𝐴 No 𝐴 ∈ V))
3 elex 3400 . . 3 (𝐵𝑊𝐵 ∈ V)
43anim2i 611 . 2 ((𝐵 No 𝐵𝑊) → (𝐵 No 𝐵 ∈ V))
5 id 22 . 2 (∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏 → ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏)
6 simp1l 1255 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝐴 No )
7 simp1r 1256 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝐴 ∈ V)
8 simp2r 1258 . . . 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 32376 . . . 4 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )
126, 7, 8, 11syl3anc 1491 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → 𝑍 No )
13 simplll 792 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝐴 No )
14 simpllr 794 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝐴 ∈ V)
15 simplrr 797 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝐵 ∈ V)
16 simpr 478 . . . . . 6 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝑎𝐴)
179, 10noetalem2 32377 . . . . . 6 (((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑎𝐴) → 𝑎 <s 𝑍)
1813, 14, 15, 16, 17syl31anc 1493 . . . . 5 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) ∧ 𝑎𝐴) → 𝑎 <s 𝑍)
1918ralrimiva 3147 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ∀𝑎𝐴 𝑎 <s 𝑍)
20193adant3 1163 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∀𝑎𝐴 𝑎 <s 𝑍)
219, 10noetalem3 32378 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ∀𝑏𝐵 𝑍 <s 𝑏)
229, 10noetalem4 32379 . . . 4 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V)) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))
23223adant3 1163 . . 3 (((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵)))
24 breq2 4847 . . . . . 6 (𝑧 = 𝑍 → (𝑎 <s 𝑧𝑎 <s 𝑍))
2524ralbidv 3167 . . . . 5 (𝑧 = 𝑍 → (∀𝑎𝐴 𝑎 <s 𝑧 ↔ ∀𝑎𝐴 𝑎 <s 𝑍))
26 breq1 4846 . . . . . 6 (𝑧 = 𝑍 → (𝑧 <s 𝑏𝑍 <s 𝑏))
2726ralbidv 3167 . . . . 5 (𝑧 = 𝑍 → (∀𝑏𝐵 𝑧 <s 𝑏 ↔ ∀𝑏𝐵 𝑍 <s 𝑏))
28 fveq2 6411 . . . . . 6 (𝑧 = 𝑍 → ( bday 𝑧) = ( bday 𝑍))
2928sseq1d 3828 . . . . 5 (𝑧 = 𝑍 → (( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵)) ↔ ( bday 𝑍) ⊆ suc ( bday “ (𝐴𝐵))))
3025, 27, 293anbi123d 1561 . . . 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 3497 . . 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 1492 . 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 1200 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 385  w3a 1108   = wceq 1653  wcel 2157  {cab 2785  wral 3089  wrex 3090  Vcvv 3385  cdif 3766  cun 3767  wss 3769  ifcif 4277  {csn 4368  cop 4374   cuni 4628   class class class wbr 4843  cmpt 4922   × cxp 5310  dom cdm 5312  cres 5314  cima 5315  suc csuc 5943  cio 6062  cfv 6101  crio 6838  1𝑜c1o 7792  2𝑜c2o 7793   No csur 32306   <s cslt 32307   bday cbday 32308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-1o 7799  df-2o 7800  df-no 32309  df-slt 32310  df-bday 32311
This theorem is referenced by:  noeta  32381
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