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Theorem onsetrec 49227
Description: Construct On using set recursion. When 𝑥 ∈ On, the function 𝐹 constructs the least ordinal greater than any of the elements of 𝑥, which is 𝑥 for a limit ordinal and suc 𝑥 for a successor ordinal.

For example, (𝐹‘{1o, 2o}) = { {1o, 2o}, suc {1o, 2o}} = {2o, 3o} which contains 3o, and (𝐹‘ω) = { ω, suc ω} = {ω, ω +o 1o}, which contains ω. If we start with the empty set and keep applying 𝐹 transfinitely many times, all ordinal numbers will be generated.

Any function 𝐹 fulfilling lemmas onsetreclem2 49225 and onsetreclem3 49226 will recursively generate On; for example, 𝐹 = (𝑥 ∈ V ↦ suc suc 𝑥}) also works. Whether this function or the function in the theorem is used, taking this theorem as a definition of On is unsatisfying because it relies on the different properties of limit and successor ordinals. A different approach could be to let 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝒫 𝑥 ∣ Tr 𝑦}), based on dfon2 35793.

The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable condition between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 22-Jun-2021.)

Hypothesis
Ref Expression
onsetrec.1 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
Assertion
Ref Expression
onsetrec setrecs(𝐹) = On

Proof of Theorem onsetrec
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 setrecs(𝐹) = setrecs(𝐹)
2 onsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
32onsetreclem2 49225 . . . . . 6 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
43ax-gen 1795 . . . . 5 𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
54a1i 11 . . . 4 (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On))
61, 5setrec2v 49215 . . 3 (⊤ → setrecs(𝐹) ⊆ On)
76mptru 1547 . 2 setrecs(𝐹) ⊆ On
8 vex 3484 . . . . . . 7 𝑎 ∈ V
98a1i 11 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V)
10 id 22 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹))
111, 9, 10setrec1 49210 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
122onsetreclem3 49226 . . . . 5 (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
13 ssel 3977 . . . . 5 ((𝐹𝑎) ⊆ setrecs(𝐹) → (𝑎 ∈ (𝐹𝑎) → 𝑎 ∈ setrecs(𝐹)))
1411, 12, 13syl2im 40 . . . 4 (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ On → 𝑎 ∈ setrecs(𝐹)))
1514com12 32 . . 3 (𝑎 ∈ On → (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)))
1615rgen 3063 . 2 𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
17 tfi 7874 . 2 ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On)
187, 16, 17mp2an 692 1 setrecs(𝐹) = On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  wtru 1541  wcel 2108  wral 3061  Vcvv 3480  wss 3951  {cpr 4628   cuni 4907  cmpt 5225  Oncon0 6384  suc csuc 6386  cfv 6561  setrecscsetrecs 49202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-r1 9804  df-rank 9805  df-setrecs 49203
This theorem is referenced by: (None)
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