Theorem noeta 27728

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   𝑦,𝐵,𝑧   𝑧,𝑂   𝑥,𝐵

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 𝑗 ℎ 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 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 27687	. 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 2737	. . 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 27702	. 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 27727	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 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∪ cun 3901   ⊆ wss 3903  ifcif 4481  {csn 4582  ⟨cop 4588   class class class wbr 5100   ↦ cmpt 5181  dom cdm 5634   ↾ cres 5636   “ cima 5637  Oncon0 6327  suc csuc 6329  ℩cio 6456  ‘cfv 6502  ℩crio 7326  1_oc1o 8402  2_oc2o 8403   No csur 27624   <s clts 27625   bday cbday 27626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  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 6330  df-on 6331  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-1o 8409  df-2o 8410  df-no 27627  df-lts 27628  df-bday 27629

This theorem is referenced by:  noeta2  27774  etaslts  27806