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Theorem noeta 31993
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, there is an upper bound on the birthday of that surreal. Alling's axiom FE. (Contributed by Scott Fenton, 7-Dec-2021.)
Assertion
Ref Expression
noeta (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐴 𝑥 <s 𝑧 ∧ ∀𝑦𝐵 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)

Proof of Theorem noeta
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2718 . . . . . . 7 (𝑓 = 𝑐 → (𝑓 ∈ dom 𝑑𝑐 ∈ dom 𝑑))
2 suceq 5828 . . . . . . . . . . 11 (𝑓 = 𝑐 → suc 𝑓 = suc 𝑐)
32reseq2d 5428 . . . . . . . . . 10 (𝑓 = 𝑐 → (𝑑 ↾ suc 𝑓) = (𝑑 ↾ suc 𝑐))
42reseq2d 5428 . . . . . . . . . 10 (𝑓 = 𝑐 → (𝑒 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑐))
53, 4eqeq12d 2666 . . . . . . . . 9 (𝑓 = 𝑐 → ((𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓) ↔ (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)))
65imbi2d 329 . . . . . . . 8 (𝑓 = 𝑐 → ((¬ 𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ↔ (¬ 𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))))
76ralbidv 3015 . . . . . . 7 (𝑓 = 𝑐 → (∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ↔ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))))
8 fveq2 6229 . . . . . . . 8 (𝑓 = 𝑐 → (𝑑𝑓) = (𝑑𝑐))
98eqeq1d 2653 . . . . . . 7 (𝑓 = 𝑐 → ((𝑑𝑓) = 𝑎 ↔ (𝑑𝑐) = 𝑎))
101, 7, 93anbi123d 1439 . . . . . 6 (𝑓 = 𝑐 → ((𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎) ↔ (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎)))
1110rexbidv 3081 . . . . 5 (𝑓 = 𝑐 → (∃𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎) ↔ ∃𝑑𝐴 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎)))
1211iotabidv 5910 . . . 4 (𝑓 = 𝑐 → (℩𝑎𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎)) = (℩𝑎𝑑𝐴 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑𝑐) = 𝑎)))
1312cbvmptv 4783 . . 3 (𝑓 ∈ {𝑏 ∣ ∃𝑑𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎))) = (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)))
14 ifeq2 4124 . . 3 ((𝑓 ∈ {𝑏 ∣ ∃𝑑𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎))) = (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))) → if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2𝑜⟩}), (𝑓 ∈ {𝑏 ∣ ∃𝑑𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎)))) = 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)))))
1513, 14ax-mp 5 . 2 if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2𝑜⟩}), (𝑓 ∈ {𝑏 ∣ ∃𝑑𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎)))) = 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))))
16 eqid 2651 . 2 (if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2𝑜⟩}), (𝑓 ∈ {𝑏 ∣ ∃𝑑𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎)))) ∪ ((suc ( bday 𝐵) ∖ dom if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2𝑜⟩}), (𝑓 ∈ {𝑏 ∣ ∃𝑑𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎))))) × {1𝑜})) = (if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2𝑜⟩}), (𝑓 ∈ {𝑏 ∣ ∃𝑑𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎)))) ∪ ((suc ( bday 𝐵) ∖ dom if(∃𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏, ((𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (𝑎𝐴𝑏𝐴 ¬ 𝑎 <s 𝑏), 2𝑜⟩}), (𝑓 ∈ {𝑏 ∣ ∃𝑑𝐴 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎𝑑𝐴 (𝑓 ∈ dom 𝑑 ∧ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ∧ (𝑑𝑓) = 𝑎))))) × {1𝑜}))
1715, 16noetalem5 31992 1 (((𝐴 No 𝐴𝑉) ∧ (𝐵 No 𝐵𝑊) ∧ ∀𝑥𝐴𝑦𝐵 𝑥 <s 𝑦) → ∃𝑧 No (∀𝑥𝐴 𝑥 <s 𝑧 ∧ ∀𝑦𝐵 𝑧 <s 𝑦 ∧ ( bday 𝑧) ⊆ suc ( bday “ (𝐴𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  cdif 3604  cun 3605  wss 3607  ifcif 4119  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  cres 5145  cima 5146  suc csuc 5763  cio 5887  cfv 5926  crio 6650  1𝑜c1o 7598  2𝑜c2o 7599   No csur 31918   <s cslt 31919   bday cbday 31920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-1o 7605  df-2o 7606  df-no 31921  df-slt 31922  df-bday 31923
This theorem is referenced by:  conway  32035  etasslt  32045
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