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Theorem noetalem2 27630
Description: Lemma for noeta 27631. The full statement of the theorem with hypotheses in place. (Contributed by Scott Fenton, 10-Aug-2024.)
Hypotheses
Ref Expression
noetalem2.1 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))))
noetalem2.2 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
Assertion
Ref Expression
noetalem2 ((((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂)) → ∃𝑐 No (∀𝑎𝐴 𝑎 <s 𝑐 ∧ ∀𝑏𝐵 𝑐 <s 𝑏 ∧ ( bday 𝑐) ⊆ 𝑂))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑐   𝑔,𝑎,𝐴,𝑢,𝑣,𝑥,𝑦   𝐵,𝑎,𝑏,𝑐   𝑔,𝑏,𝐵   𝑢,𝐵   𝑣,𝑏,𝐵,𝑥,𝑦   𝑢,𝑔,𝑣,𝑥,𝑦   𝑂,𝑐   𝑢,𝑂,𝑦   𝑆,𝑎,𝑏,𝑐   𝑆,𝑔,𝑥   𝑇,𝑎,𝑏,𝑐   𝑇,𝑔,𝑥   𝑣,𝑢,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑦,𝑣,𝑢)   𝑇(𝑦,𝑣,𝑢)   𝑂(𝑥,𝑣,𝑔,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑣,𝑢,𝑔,𝑎,𝑏,𝑐)   𝑊(𝑥,𝑦,𝑣,𝑢,𝑔,𝑎,𝑏,𝑐)

Proof of Theorem noetalem2
StepHypRef Expression
1 elex 3487 . . . 4 (𝐴𝑉𝐴 ∈ V)
21anim2i 616 . . 3 ((𝐴 No 𝐴𝑉) → (𝐴 No 𝐴 ∈ V))
3 elex 3487 . . . 4 (𝐵𝑊𝐵 ∈ V)
43anim2i 616 . . 3 ((𝐵 No 𝐵𝑊) → (𝐵 No 𝐵 ∈ V))
5 id 22 . . 3 (∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏 → ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏)
62, 4, 53anim123i 1148 . 2 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) → ((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏))
7 noetalem2.1 . . . 4 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))))
8 noetalem2.2 . . . 4 𝑇 = if(∃𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥, ((𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (𝑥𝐵𝑦𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥𝑢𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣𝐵𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢𝑔) = 𝑥))))
9 eqid 2726 . . . 4 (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o})) = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))
10 eqid 2726 . . . 4 (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) = (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}))
117, 8, 9, 10noetalem1 27629 . . 3 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂)) → ((𝑆 No ∧ (∀𝑎𝐴 𝑎 <s 𝑆 ∧ ∀𝑏𝐵 𝑆 <s 𝑏 ∧ ( bday 𝑆) ⊆ 𝑂)) ∨ (𝑇 No ∧ (∀𝑎𝐴 𝑎 <s 𝑇 ∧ ∀𝑏𝐵 𝑇 <s 𝑏 ∧ ( bday 𝑇) ⊆ 𝑂))))
12 breq2 5145 . . . . . . 7 (𝑐 = 𝑆 → (𝑎 <s 𝑐𝑎 <s 𝑆))
1312ralbidv 3171 . . . . . 6 (𝑐 = 𝑆 → (∀𝑎𝐴 𝑎 <s 𝑐 ↔ ∀𝑎𝐴 𝑎 <s 𝑆))
14 breq1 5144 . . . . . . 7 (𝑐 = 𝑆 → (𝑐 <s 𝑏𝑆 <s 𝑏))
1514ralbidv 3171 . . . . . 6 (𝑐 = 𝑆 → (∀𝑏𝐵 𝑐 <s 𝑏 ↔ ∀𝑏𝐵 𝑆 <s 𝑏))
16 fveq2 6885 . . . . . . 7 (𝑐 = 𝑆 → ( bday 𝑐) = ( bday 𝑆))
1716sseq1d 4008 . . . . . 6 (𝑐 = 𝑆 → (( bday 𝑐) ⊆ 𝑂 ↔ ( bday 𝑆) ⊆ 𝑂))
1813, 15, 173anbi123d 1432 . . . . 5 (𝑐 = 𝑆 → ((∀𝑎𝐴 𝑎 <s 𝑐 ∧ ∀𝑏𝐵 𝑐 <s 𝑏 ∧ ( bday 𝑐) ⊆ 𝑂) ↔ (∀𝑎𝐴 𝑎 <s 𝑆 ∧ ∀𝑏𝐵 𝑆 <s 𝑏 ∧ ( bday 𝑆) ⊆ 𝑂)))
1918rspcev 3606 . . . 4 ((𝑆 No ∧ (∀𝑎𝐴 𝑎 <s 𝑆 ∧ ∀𝑏𝐵 𝑆 <s 𝑏 ∧ ( bday 𝑆) ⊆ 𝑂)) → ∃𝑐 No (∀𝑎𝐴 𝑎 <s 𝑐 ∧ ∀𝑏𝐵 𝑐 <s 𝑏 ∧ ( bday 𝑐) ⊆ 𝑂))
20 breq2 5145 . . . . . . 7 (𝑐 = 𝑇 → (𝑎 <s 𝑐𝑎 <s 𝑇))
2120ralbidv 3171 . . . . . 6 (𝑐 = 𝑇 → (∀𝑎𝐴 𝑎 <s 𝑐 ↔ ∀𝑎𝐴 𝑎 <s 𝑇))
22 breq1 5144 . . . . . . 7 (𝑐 = 𝑇 → (𝑐 <s 𝑏𝑇 <s 𝑏))
2322ralbidv 3171 . . . . . 6 (𝑐 = 𝑇 → (∀𝑏𝐵 𝑐 <s 𝑏 ↔ ∀𝑏𝐵 𝑇 <s 𝑏))
24 fveq2 6885 . . . . . . 7 (𝑐 = 𝑇 → ( bday 𝑐) = ( bday 𝑇))
2524sseq1d 4008 . . . . . 6 (𝑐 = 𝑇 → (( bday 𝑐) ⊆ 𝑂 ↔ ( bday 𝑇) ⊆ 𝑂))
2621, 23, 253anbi123d 1432 . . . . 5 (𝑐 = 𝑇 → ((∀𝑎𝐴 𝑎 <s 𝑐 ∧ ∀𝑏𝐵 𝑐 <s 𝑏 ∧ ( bday 𝑐) ⊆ 𝑂) ↔ (∀𝑎𝐴 𝑎 <s 𝑇 ∧ ∀𝑏𝐵 𝑇 <s 𝑏 ∧ ( bday 𝑇) ⊆ 𝑂)))
2726rspcev 3606 . . . 4 ((𝑇 No ∧ (∀𝑎𝐴 𝑎 <s 𝑇 ∧ ∀𝑏𝐵 𝑇 <s 𝑏 ∧ ( bday 𝑇) ⊆ 𝑂)) → ∃𝑐 No (∀𝑎𝐴 𝑎 <s 𝑐 ∧ ∀𝑏𝐵 𝑐 <s 𝑏 ∧ ( bday 𝑐) ⊆ 𝑂))
2819, 27jaoi 854 . . 3 (((𝑆 No ∧ (∀𝑎𝐴 𝑎 <s 𝑆 ∧ ∀𝑏𝐵 𝑆 <s 𝑏 ∧ ( bday 𝑆) ⊆ 𝑂)) ∨ (𝑇 No ∧ (∀𝑎𝐴 𝑎 <s 𝑇 ∧ ∀𝑏𝐵 𝑇 <s 𝑏 ∧ ( bday 𝑇) ⊆ 𝑂))) → ∃𝑐 No (∀𝑎𝐴 𝑎 <s 𝑐 ∧ ∀𝑏𝐵 𝑐 <s 𝑏 ∧ ( bday 𝑐) ⊆ 𝑂))
2911, 28syl 17 . 2 ((((𝐴 No 𝐴 ∈ V) ∧ (𝐵 No 𝐵 ∈ V) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂)) → ∃𝑐 No (∀𝑎𝐴 𝑎 <s 𝑐 ∧ ∀𝑏𝐵 𝑐 <s 𝑏 ∧ ( bday 𝑐) ⊆ 𝑂))
306, 29sylan 579 1 ((((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂)) → ∃𝑐 No (∀𝑎𝐴 𝑎 <s 𝑐 ∧ ∀𝑏𝐵 𝑐 <s 𝑏 ∧ ( bday 𝑐) ⊆ 𝑂))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  {cab 2703  wral 3055  wrex 3064  Vcvv 3468  cdif 3940  cun 3941  wss 3943  ifcif 4523  {csn 4623  cop 4629   cuni 4902   class class class wbr 5141  cmpt 5224   × cxp 5667  dom cdm 5669  cres 5671  cima 5672  Oncon0 6358  suc csuc 6360  cio 6487  cfv 6537  crio 7360  1oc1o 8460  2oc2o 8461   No csur 27528   <s cslt 27529   bday cbday 27530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-1o 8467  df-2o 8468  df-no 27531  df-slt 27532  df-bday 27533
This theorem is referenced by:  noeta  27631
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