Theorem noeta 27806

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 2740	. . 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 27765	. 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 2740	. . 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 27780	. 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 27805	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 1087   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076   ∪ cun 3974   ⊆ wss 3976  ifcif 4548  {csn 4648  ⟨cop 4654   class class class wbr 5166   ↦ cmpt 5249  dom cdm 5700   ↾ cres 5702   “ cima 5703  Oncon0 6395  suc csuc 6397  ℩cio 6523  ‘cfv 6573  ℩crio 7403  1_oc1o 8515  2_oc2o 8516   No csur 27702   <s cslt 27703   bday cbday 27704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707

This theorem is referenced by:  noeta2  27847  etasslt  27876