MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  noeta Structured version   Visualization version   GIF version

Theorem noeta 27107
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 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂)) → ∃𝑧 No (∀𝑥𝐴 𝑥 <s 𝑧 ∧ ∀𝑦𝐵 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 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 if(∃𝑓𝐴𝑔𝐴 ¬ 𝑓 <s 𝑔, ((𝑓𝐴𝑔𝐴 ¬ 𝑓 <s 𝑔) ∪ {⟨dom (𝑓𝐴𝑔𝐴 ¬ 𝑓 <s 𝑔), 2o⟩}), ( ∈ {𝑔 ∣ ∃𝑗𝐴 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘𝐴𝑘 <s 𝑗 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓𝑗𝐴 ( ∈ dom 𝑗 ∧ ∀𝑘𝐴𝑘 <s 𝑗 → (𝑗 ↾ suc ) = (𝑘 ↾ suc )) ∧ (𝑗) = 𝑓)))) = if(∃𝑓𝐴𝑔𝐴 ¬ 𝑓 <s 𝑔, ((𝑓𝐴𝑔𝐴 ¬ 𝑓 <s 𝑔) ∪ {⟨dom (𝑓𝐴𝑔𝐴 ¬ 𝑓 <s 𝑔), 2o⟩}), ( ∈ {𝑔 ∣ ∃𝑗𝐴 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘𝐴𝑘 <s 𝑗 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓𝑗𝐴 ( ∈ dom 𝑗 ∧ ∀𝑘𝐴𝑘 <s 𝑗 → (𝑗 ↾ suc ) = (𝑘 ↾ suc )) ∧ (𝑗) = 𝑓))))
21nosupcbv 27066 . 2 if(∃𝑓𝐴𝑔𝐴 ¬ 𝑓 <s 𝑔, ((𝑓𝐴𝑔𝐴 ¬ 𝑓 <s 𝑔) ∪ {⟨dom (𝑓𝐴𝑔𝐴 ¬ 𝑓 <s 𝑔), 2o⟩}), ( ∈ {𝑔 ∣ ∃𝑗𝐴 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘𝐴𝑘 <s 𝑗 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓𝑗𝐴 ( ∈ dom 𝑗 ∧ ∀𝑘𝐴𝑘 <s 𝑗 → (𝑗 ↾ suc ) = (𝑘 ↾ suc )) ∧ (𝑗) = 𝑓)))) = if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐴 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎))))
3 eqid 2733 . . 3 if(∃𝑓𝐵𝑔𝐵 ¬ 𝑔 <s 𝑓, ((𝑓𝐵𝑔𝐵 ¬ 𝑔 <s 𝑓) ∪ {⟨dom (𝑓𝐵𝑔𝐵 ¬ 𝑔 <s 𝑓), 1o⟩}), ( ∈ {𝑔 ∣ ∃𝑗𝐵 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘𝐵𝑗 <s 𝑘 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓𝑗𝐵 ( ∈ dom 𝑗 ∧ ∀𝑘𝐵𝑗 <s 𝑘 → (𝑗 ↾ suc ) = (𝑘 ↾ suc )) ∧ (𝑗) = 𝑓)))) = if(∃𝑓𝐵𝑔𝐵 ¬ 𝑔 <s 𝑓, ((𝑓𝐵𝑔𝐵 ¬ 𝑔 <s 𝑓) ∪ {⟨dom (𝑓𝐵𝑔𝐵 ¬ 𝑔 <s 𝑓), 1o⟩}), ( ∈ {𝑔 ∣ ∃𝑗𝐵 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘𝐵𝑗 <s 𝑘 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓𝑗𝐵 ( ∈ dom 𝑗 ∧ ∀𝑘𝐵𝑗 <s 𝑘 → (𝑗 ↾ suc ) = (𝑘 ↾ suc )) ∧ (𝑗) = 𝑓))))
43noinfcbv 27081 . 2 if(∃𝑓𝐵𝑔𝐵 ¬ 𝑔 <s 𝑓, ((𝑓𝐵𝑔𝐵 ¬ 𝑔 <s 𝑓) ∪ {⟨dom (𝑓𝐵𝑔𝐵 ¬ 𝑔 <s 𝑓), 1o⟩}), ( ∈ {𝑔 ∣ ∃𝑗𝐵 (𝑔 ∈ dom 𝑗 ∧ ∀𝑘𝐵𝑗 <s 𝑘 → (𝑗 ↾ suc 𝑔) = (𝑘 ↾ suc 𝑔)))} ↦ (℩𝑓𝑗𝐵 ( ∈ dom 𝑗 ∧ ∀𝑘𝐵𝑗 <s 𝑘 → (𝑗 ↾ suc ) = (𝑘 ↾ suc )) ∧ (𝑗) = 𝑓)))) = if(∃𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎, ((𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (𝑎𝐵𝑏𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐵𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎))))
52, 4noetalem2 27106 1 ((((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴𝐵)) ⊆ 𝑂)) → ∃𝑧 No (∀𝑥𝐴 𝑥 <s 𝑧 ∧ ∀𝑦𝐵 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ 𝑂))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  cun 3909  wss 3911  ifcif 4487  {csn 4587  cop 4593   class class class wbr 5106  cmpt 5189  dom cdm 5634  cres 5636  cima 5637  Oncon0 6318  suc csuc 6320  cio 6447  cfv 6497  crio 7313  1oc1o 8406  2oc2o 8407   No csur 27004   <s cslt 27005   bday cbday 27006
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  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 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-1o 8413  df-2o 8414  df-no 27007  df-slt 27008  df-bday 27009
This theorem is referenced by:  noeta2  27146  etasslt  27174
  Copyright terms: Public domain W3C validator