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Theorem onsetrec 42964
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 42962 and onsetreclem3 42963 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 32002.

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 2760 . . . 4 setrecs(𝐹) = setrecs(𝐹)
2 onsetrec.1 . . . . . . 7 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
32onsetreclem2 42962 . . . . . 6 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
43ax-gen 1871 . . . . 5 𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
54a1i 11 . . . 4 (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹𝑎) ⊆ On))
61, 5setrec2v 42953 . . 3 (⊤ → setrecs(𝐹) ⊆ On)
76trud 1642 . 2 setrecs(𝐹) ⊆ On
8 vex 3343 . . . . . . 7 𝑎 ∈ V
98a1i 11 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V)
10 id 22 . . . . . 6 (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹))
111, 9, 10setrec1 42948 . . . . 5 (𝑎 ⊆ setrecs(𝐹) → (𝐹𝑎) ⊆ setrecs(𝐹))
122onsetreclem3 42963 . . . . 5 (𝑎 ∈ On → 𝑎 ∈ (𝐹𝑎))
13 ssel 3738 . . . . 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 7218 . 2 ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On)
187, 16, 17mp2an 710 1 setrecs(𝐹) = On
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1630   = wceq 1632  wtru 1633  wcel 2139  wral 3050  Vcvv 3340  wss 3715  {cpr 4323   cuni 4588  cmpt 4881  Oncon0 5884  suc csuc 5886  cfv 6049  setrecscsetrecs 42940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-reg 8662  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800  df-rank 8801  df-setrecs 42941
This theorem is referenced by: (None)
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