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Theorem onsetrec 45163
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 45161 and onsetreclem3 45162 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 33094.

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 2824 . . . 4 setrecs(𝐹) = setrecs(𝐹)
2 onsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
32onsetreclem2 45161 . . . . . 6 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
43ax-gen 1797 . . . . 5 𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
54a1i 11 . . . 4 (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On))
61, 5setrec2v 45152 . . 3 (⊤ → setrecs(𝐹) ⊆ On)
76mptru 1545 . 2 setrecs(𝐹) ⊆ On
8 vex 3483 . . . . . . 7 𝑎 ∈ V
98a1i 11 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V)
10 id 22 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹))
111, 9, 10setrec1 45147 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
122onsetreclem3 45162 . . . . 5 (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
13 ssel 3946 . . . . 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 3143 . 2 𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
17 tfi 7562 . 2 ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On)
187, 16, 17mp2an 691 1 setrecs(𝐹) = On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536   = wceq 1538  wtru 1539  wcel 2115  wral 3133  Vcvv 3480  wss 3919  {cpr 4552   cuni 4824  cmpt 5132  Oncon0 6178  suc csuc 6180  cfv 6343  setrecscsetrecs 45139
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-reg 9053  ax-inf2 9101
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-r1 9190  df-rank 9191  df-setrecs 45140
This theorem is referenced by: (None)
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