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Theorem noetalem1 31988
 Description: Lemma for noeta 31993. Establish that our final surreal really is a surreal. (Contributed by Scott Fenton, 6-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
noetalem1 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )
Distinct variable group:   𝐴,𝑔,𝑢,𝑣,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑆(𝑥,𝑦,𝑣,𝑢,𝑔)   𝑍(𝑥,𝑦,𝑣,𝑢,𝑔)

Proof of Theorem noetalem1
StepHypRef Expression
1 noetalem.2 . 2 𝑍 = (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜}))
2 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))))
32nosupno 31974 . . . 4 ((𝐴 No 𝐴 ∈ V) → 𝑆 No )
433adant3 1101 . . 3 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑆 No )
5 bdayimaon 31968 . . . 4 (𝐵 ∈ V → suc ( bday 𝐵) ∈ On)
653ad2ant3 1104 . . 3 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → suc ( bday 𝐵) ∈ On)
7 1on 7612 . . . . . 6 1𝑜 ∈ On
87elexi 3244 . . . . 5 1𝑜 ∈ V
98prid1 4329 . . . 4 1𝑜 ∈ {1𝑜, 2𝑜}
109noextendseq 31945 . . 3 ((𝑆 No ∧ suc ( bday 𝐵) ∈ On) → (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) ∈ No )
114, 6, 10syl2anc 694 . 2 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑆 ∪ ((suc ( bday 𝐵) ∖ dom 𝑆) × {1𝑜})) ∈ No )
121, 11syl5eqel 2734 1 ((𝐴 No 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 No )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ⊆ wss 3607  ifcif 4119  {csn 4210  ⟨cop 4216  ∪ cuni 4468   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  dom cdm 5143   ↾ cres 5145   “ cima 5146  Oncon0 5761  suc csuc 5763  ℩cio 5887  ‘cfv 5926  ℩crio 6650  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918
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