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Theorem onsetrec 48235
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 48233 and onsetreclem3 48234 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 35429.

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 2728 . . . 4 setrecs(𝐹) = setrecs(𝐹)
2 onsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
32onsetreclem2 48233 . . . . . 6 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
43ax-gen 1789 . . . . 5 𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
54a1i 11 . . . 4 (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On))
61, 5setrec2v 48223 . . 3 (⊤ → setrecs(𝐹) ⊆ On)
76mptru 1540 . 2 setrecs(𝐹) ⊆ On
8 vex 3477 . . . . . . 7 𝑎 ∈ V
98a1i 11 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V)
10 id 22 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹))
111, 9, 10setrec1 48218 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
122onsetreclem3 48234 . . . . 5 (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
13 ssel 3975 . . . . 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 3060 . 2 𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))
17 tfi 7865 . 2 ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On)
187, 16, 17mp2an 690 1 setrecs(𝐹) = On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531   = wceq 1533  wtru 1534  wcel 2098  wral 3058  Vcvv 3473  wss 3949  {cpr 4634   cuni 4912  cmpt 5235  Oncon0 6374  suc csuc 6376  cfv 6553  setrecscsetrecs 48210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-reg 9625  ax-inf2 9674
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7879  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-r1 9797  df-rank 9798  df-setrecs 48211
This theorem is referenced by: (None)
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