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Theorem onsetrec 42216
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 42214 and onsetreclem3 42215 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 31671.

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 2620 . . . 4 setrecs(𝐹) = setrecs(𝐹)
2 onsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
32onsetreclem2 42214 . . . . . 6 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
43ax-gen 1720 . . . . 5 𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
54a1i 11 . . . 4 (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On))
61, 5setrec2v 42208 . . 3 (⊤ → setrecs(𝐹) ⊆ On)
76trud 1491 . 2 setrecs(𝐹) ⊆ On
8 vex 3198 . . . . . . 7 𝑎 ∈ V
98a1i 11 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V)
10 id 22 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹))
111, 9, 10setrec1 42203 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
122onsetreclem3 42215 . . . . 5 (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
13 ssel 3589 . . . . 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 2919 . 2 𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
17 tfi 7038 . 2 ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On)
187, 16, 17mp2an 707 1 setrecs(𝐹) = On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1479   = wceq 1481  wtru 1482  wcel 1988  wral 2909  Vcvv 3195  wss 3567  {cpr 4170   cuni 4427  cmpt 4720  Oncon0 5711  suc csuc 5713  cfv 5876  setrecscsetrecs 42195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-reg 8482  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-r1 8612  df-rank 8613  df-setrecs 42196
This theorem is referenced by: (None)
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