Theorem noeta 27682

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 2731	. . 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 27641	. 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 2731	. . 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 27656	. 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 27681	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 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056   ∪ cun 3895   ⊆ wss 3897  ifcif 4472  {csn 4573  ⟨cop 4579   class class class wbr 5089   ↦ cmpt 5170  dom cdm 5614   ↾ cres 5616   “ cima 5617  Oncon0 6306  suc csuc 6308  ℩cio 6435  ‘cfv 6481  ℩crio 7302  1_oc1o 8378  2_oc2o 8379   No csur 27578   <s cslt 27579   bday cbday 27580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-1o 8385  df-2o 8386  df-no 27581  df-slt 27582  df-bday 27583

This theorem is referenced by:  noeta2  27724  etasslt  27754