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Theorem noetalem2 27106
Description: Lemma for noeta 27107. 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 3462 . . . 4 (𝐴𝑉𝐴 ∈ V)
21anim2i 618 . . 3 ((𝐴 No 𝐴𝑉) → (𝐴 No 𝐴 ∈ V))
3 elex 3462 . . . 4 (𝐵𝑊𝐵 ∈ V)
43anim2i 618 . . 3 ((𝐵 No 𝐵𝑊) → (𝐵 No 𝐵 ∈ V))
5 id 22 . . 3 (∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏 → ∀𝑎𝐴𝑏𝐵 𝑎 <s 𝑏)
62, 4, 53anim123i 1152 . 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 2733 . . . 4 (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o})) = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1o}))
10 eqid 2733 . . . 4 (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) = (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}))
117, 8, 9, 10noetalem1 27105 . . 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 5110 . . . . . . 7 (𝑐 = 𝑆 → (𝑎 <s 𝑐𝑎 <s 𝑆))
1312ralbidv 3171 . . . . . 6 (𝑐 = 𝑆 → (∀𝑎𝐴 𝑎 <s 𝑐 ↔ ∀𝑎𝐴 𝑎 <s 𝑆))
14 breq1 5109 . . . . . . 7 (𝑐 = 𝑆 → (𝑐 <s 𝑏𝑆 <s 𝑏))
1514ralbidv 3171 . . . . . 6 (𝑐 = 𝑆 → (∀𝑏𝐵 𝑐 <s 𝑏 ↔ ∀𝑏𝐵 𝑆 <s 𝑏))
16 fveq2 6843 . . . . . . 7 (𝑐 = 𝑆 → ( bday 𝑐) = ( bday 𝑆))
1716sseq1d 3976 . . . . . 6 (𝑐 = 𝑆 → (( bday 𝑐) ⊆ 𝑂 ↔ ( bday 𝑆) ⊆ 𝑂))
1813, 15, 173anbi123d 1437 . . . . 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 3580 . . . 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 5110 . . . . . . 7 (𝑐 = 𝑇 → (𝑎 <s 𝑐𝑎 <s 𝑇))
2120ralbidv 3171 . . . . . 6 (𝑐 = 𝑇 → (∀𝑎𝐴 𝑎 <s 𝑐 ↔ ∀𝑎𝐴 𝑎 <s 𝑇))
22 breq1 5109 . . . . . . 7 (𝑐 = 𝑇 → (𝑐 <s 𝑏𝑇 <s 𝑏))
2322ralbidv 3171 . . . . . 6 (𝑐 = 𝑇 → (∀𝑏𝐵 𝑐 <s 𝑏 ↔ ∀𝑏𝐵 𝑇 <s 𝑏))
24 fveq2 6843 . . . . . . 7 (𝑐 = 𝑇 → ( bday 𝑐) = ( bday 𝑇))
2524sseq1d 3976 . . . . . 6 (𝑐 = 𝑇 → (( bday 𝑐) ⊆ 𝑂 ↔ ( bday 𝑇) ⊆ 𝑂))
2621, 23, 253anbi123d 1437 . . . . 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 3580 . . . 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 856 . . 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 581 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  Vcvv 3444  cdif 3908  cun 3909  wss 3911  ifcif 4487  {csn 4587  cop 4593   cuni 4866   class class class wbr 5106  cmpt 5189   × cxp 5632  dom cdm 5634  cres 5636  cima 5637  Oncon0 6318  suc csuc 6320  cio 6447  cfv 6497  crio 7313  1oc1o 8406  2oc2o 8407   No csur 27004   <s cslt 27005   bday cbday 27006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009
This theorem is referenced by:  noeta  27107
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