Theorem noeta 27803

Description: The full-eta axiom for the surreal numbers. This is the single most important property of the surreals. It says that, given two sets of surreals such that one comes completely before the other, there is a surreal lying strictly between the two. Furthermore, if the birthdays of members of 𝐴 and 𝐵 are strictly bounded above by 𝑂, then 𝑂 non-strictly bounds the separator. Axiom FE of [Alling] p. 185. (Contributed by Scott Fenton, 9-Aug-2024.)

Assertion

Ref	Expression
noeta	⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday ‘𝑧) ⊆ 𝑂))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑧,𝑂   𝑥,𝑦,𝑧

Allowed substitution hints:   𝑂(𝑥,𝑦)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem noeta

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ if(∃𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 ¬ 𝑓 <s 𝑔, ((℩𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 ¬ 𝑓 <s 𝑔) ∪ {⟨dom (℩𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 ¬ 𝑓 <s 𝑔), 2o⟩}), (ℎ ∈ {𝑔 ∣ ∃𝑗 ∈ 𝐴 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐴 (¬ 𝑘 <s 𝑗 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓∃𝑗 ∈ 𝐴 (ℎ ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐴 (¬ 𝑘 <s 𝑗 → (𝑗 ↾ suc ℎ) = (𝑘 ↾ suc ℎ)) ∧ (𝑗‘ℎ) = 𝑓)))) = if(∃𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 ¬ 𝑓 <s 𝑔, ((℩𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 ¬ 𝑓 <s 𝑔) ∪ {⟨dom (℩𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 ¬ 𝑓 <s 𝑔), 2o⟩}), (ℎ ∈ {𝑔 ∣ ∃𝑗 ∈ 𝐴 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐴 (¬ 𝑘 <s 𝑗 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓∃𝑗 ∈ 𝐴 (ℎ ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐴 (¬ 𝑘 <s 𝑗 → (𝑗 ↾ suc ℎ) = (𝑘 ↾ suc ℎ)) ∧ (𝑗‘ℎ) = 𝑓))))
2	1	nosupcbv 27762	. 2 ⊢ if(∃𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 ¬ 𝑓 <s 𝑔, ((℩𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 ¬ 𝑓 <s 𝑔) ∪ {⟨dom (℩𝑓 ∈ 𝐴 ∀𝑔 ∈ 𝐴 ¬ 𝑓 <s 𝑔), 2o⟩}), (ℎ ∈ {𝑔 ∣ ∃𝑗 ∈ 𝐴 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐴 (¬ 𝑘 <s 𝑗 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓∃𝑗 ∈ 𝐴 (ℎ ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐴 (¬ 𝑘 <s 𝑗 → (𝑗 ↾ suc ℎ) = (𝑘 ↾ suc ℎ)) ∧ (𝑗‘ℎ) = 𝑓)))) = if(∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏, ((℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑 ∈ 𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐴 (¬ 𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎∃𝑑 ∈ 𝐴 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐴 (¬ 𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))))
3		eqid 2735	. . 3 ⊢ if(∃𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 ¬ 𝑔 <s 𝑓, ((℩𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 ¬ 𝑔 <s 𝑓) ∪ {⟨dom (℩𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 ¬ 𝑔 <s 𝑓), 1o⟩}), (ℎ ∈ {𝑔 ∣ ∃𝑗 ∈ 𝐵 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐵 (¬ 𝑗 <s 𝑘 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓∃𝑗 ∈ 𝐵 (ℎ ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐵 (¬ 𝑗 <s 𝑘 → (𝑗 ↾ suc ℎ) = (𝑘 ↾ suc ℎ)) ∧ (𝑗‘ℎ) = 𝑓)))) = if(∃𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 ¬ 𝑔 <s 𝑓, ((℩𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 ¬ 𝑔 <s 𝑓) ∪ {⟨dom (℩𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 ¬ 𝑔 <s 𝑓), 1o⟩}), (ℎ ∈ {𝑔 ∣ ∃𝑗 ∈ 𝐵 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐵 (¬ 𝑗 <s 𝑘 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓∃𝑗 ∈ 𝐵 (ℎ ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐵 (¬ 𝑗 <s 𝑘 → (𝑗 ↾ suc ℎ) = (𝑘 ↾ suc ℎ)) ∧ (𝑗‘ℎ) = 𝑓))))
4	3	noinfcbv 27777	. 2 ⊢ if(∃𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 ¬ 𝑔 <s 𝑓, ((℩𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 ¬ 𝑔 <s 𝑓) ∪ {⟨dom (℩𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 ¬ 𝑔 <s 𝑓), 1o⟩}), (ℎ ∈ {𝑔 ∣ ∃𝑗 ∈ 𝐵 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐵 (¬ 𝑗 <s 𝑘 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓∃𝑗 ∈ 𝐵 (ℎ ∈ dom 𝑗 ∧ ∀𝑘 ∈ 𝐵 (¬ 𝑗 <s 𝑘 → (𝑗 ↾ suc ℎ) = (𝑘 ↾ suc ℎ)) ∧ (𝑗‘ℎ) = 𝑓)))) = if(∃𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎, ((℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))))
5	2, 4	noetalem2 27802	1 ⊢ ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday ‘𝑧) ⊆ 𝑂))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  ∃wrex 3068   ∪ cun 3961   ⊆ wss 3963  ifcif 4531  {csn 4631  ⟨cop 4637   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5689   ↾ cres 5691   “ cima 5692  Oncon0 6386  suc csuc 6388  ℩cio 6514  ‘cfv 6563  ℩crio 7387  1_oc1o 8498  2_oc2o 8499   No csur 27699   <s cslt 27700   bday cbday 27701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704

This theorem is referenced by:  noeta2  27844  etasslt  27873