Theorem noeta 27707

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 2736	. . 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 27666	. 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 2736	. . 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 27681	. 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 27706	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 2714  ∀wral 3051  ∃wrex 3061   ∪ cun 3887   ⊆ wss 3889  ifcif 4466  {csn 4567  ⟨cop 4573   class class class wbr 5085   ↦ cmpt 5166  dom cdm 5631   ↾ cres 5633   “ cima 5634  Oncon0 6323  suc csuc 6325  ℩cio 6452  ‘cfv 6498  ℩crio 7323  1_oc1o 8398  2_oc2o 8399   No csur 27603   <s clts 27604   bday cbday 27605

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608

This theorem is referenced by:  noeta2  27753  etaslts  27785