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Theorem noeta 33330
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 eleq1w 2875 . . . . . . 7 (𝑓 = 𝑐 → (𝑓 ∈ dom 𝑑𝑐 ∈ dom 𝑑))
2 suceq 6228 . . . . . . . . . . 11 (𝑓 = 𝑐 → suc 𝑓 = suc 𝑐)
32reseq2d 5822 . . . . . . . . . 10 (𝑓 = 𝑐 → (𝑑 ↾ suc 𝑓) = (𝑑 ↾ suc 𝑐))
42reseq2d 5822 . . . . . . . . . 10 (𝑓 = 𝑐 → (𝑒 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑐))
53, 4eqeq12d 2817 . . . . . . . . 9 (𝑓 = 𝑐 → ((𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓) ↔ (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)))
65imbi2d 344 . . . . . . . 8 (𝑓 = 𝑐 → ((¬ 𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ↔ (¬ 𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))))
76ralbidv 3165 . . . . . . 7 (𝑓 = 𝑐 → (∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑓) = (𝑒 ↾ suc 𝑓)) ↔ ∀𝑒𝐴𝑒 <s 𝑑 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐))))
8 fveqeq2 6658 . . . . . . 7 (𝑓 = 𝑐 → ((𝑑𝑓) = 𝑎 ↔ (𝑑𝑐) = 𝑎))
91, 7, 83anbi123d 1433 . . . . . 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)))
109rexbidv 3259 . . . . 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)))
1110iotabidv 6312 . . . 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)))
1211cbvmptv 5136 . . 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)))
13 ifeq2 4433 . . 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)))))
1412, 13ax-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))))
15 eqid 2801 . 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}))
1614, 15noetalem5 33329 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 399  w3a 1084   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  cdif 3881  cun 3882  wss 3884  ifcif 4428  {csn 4528  cop 4534   cuni 4803   class class class wbr 5033  cmpt 5113   × cxp 5521  dom cdm 5523  cres 5525  cima 5526  suc csuc 6165  cio 6285  cfv 6328  crio 7096  1oc1o 8082  2oc2o 8083   No csur 33255   <s cslt 33256   bday cbday 33257
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-1o 8089  df-2o 8090  df-no 33258  df-slt 33259  df-bday 33260
This theorem is referenced by:  conway  33372  etasslt  33382
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