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Theorem noetainflem1 33940
Description: Lemma for noeta 33946. Establish that this particular construction gives a surreal. (Contributed by Scott Fenton, 9-Aug-2024.)
Hypotheses
Ref Expression
noetainflem.1 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))))
noetainflem.2 𝑊 = (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}))
Assertion
Ref Expression
noetainflem1 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → 𝑊 No )
Distinct variable group:   𝐵,𝑔,𝑢,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑇(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑊(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetainflem1
StepHypRef Expression
1 noetainflem.2 . 2 𝑊 = (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o}))
2 noetainflem.1 . . . . 5 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))))
32noinfno 33921 . . . 4 ((𝐵 No 𝐵 ∈ V) → 𝑇 No )
433adant1 1129 . . 3 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → 𝑇 No )
5 bdayimaon 33896 . . . 4 (𝐴 ∈ V → suc ( bday 𝐴) ∈ On)
653ad2ant1 1132 . . 3 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → suc ( bday 𝐴) ∈ On)
7 2oex 8308 . . . . 5 2o ∈ V
87prid2 4699 . . . 4 2o ∈ {1o, 2o}
98noextendseq 33870 . . 3 ((𝑇 No ∧ suc ( bday 𝐴) ∈ On) → (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) ∈ No )
104, 6, 9syl2anc 584 . 2 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → (𝑇 ∪ ((suc ( bday 𝐴) ∖ dom 𝑇) × {2o})) ∈ No )
111, 10eqeltrid 2843 1 ((𝐴 ∈ V ∧ 𝐵 No 𝐵 ∈ V) → 𝑊 No )
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  {cab 2715  wral 3064  wrex 3065  Vcvv 3432  cdif 3884  cun 3885  wss 3887  ifcif 4459  {csn 4561  cop 4567   cuni 4839   class class class wbr 5074  cmpt 5157   × cxp 5587  dom cdm 5589  cres 5591  cima 5592  Oncon0 6266  suc csuc 6268  cio 6389  cfv 6433  crio 7231  1oc1o 8290  2oc2o 8291   No csur 33843   <s cslt 33844   bday cbday 33845
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848
This theorem is referenced by:  noetainflem3  33942  noetainflem4  33943  noetalem1  33944
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