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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetrec | Structured version Visualization version GIF version |
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 45235 and onsetreclem3 45236 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 33150. 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.) |
Ref | Expression |
---|---|
onsetrec.1 | ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) |
Ref | Expression |
---|---|
onsetrec | ⊢ setrecs(𝐹) = On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . 4 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | onsetrec.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
3 | 2 | onsetreclem2 45235 | . . . . . 6 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) |
4 | 3 | ax-gen 1797 | . . . . 5 ⊢ ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)) |
6 | 1, 5 | setrec2v 45226 | . . 3 ⊢ (⊤ → setrecs(𝐹) ⊆ On) |
7 | 6 | mptru 1545 | . 2 ⊢ setrecs(𝐹) ⊆ On |
8 | vex 3444 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V) |
10 | id 22 | . . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹)) | |
11 | 1, 9, 10 | setrec1 45221 | . . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹)) |
12 | 2 | onsetreclem3 45236 | . . . . 5 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) |
13 | ssel 3908 | . . . . 5 ⊢ ((𝐹‘𝑎) ⊆ setrecs(𝐹) → (𝑎 ∈ (𝐹‘𝑎) → 𝑎 ∈ setrecs(𝐹))) | |
14 | 11, 12, 13 | syl2im 40 | . . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ On → 𝑎 ∈ setrecs(𝐹))) |
15 | 14 | com12 32 | . . 3 ⊢ (𝑎 ∈ On → (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) |
16 | 15 | rgen 3116 | . 2 ⊢ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) |
17 | tfi 7548 | . 2 ⊢ ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On) | |
18 | 7, 16, 17 | mp2an 691 | 1 ⊢ setrecs(𝐹) = On |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 = wceq 1538 ⊤wtru 1539 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 {cpr 4527 ∪ cuni 4800 ↦ cmpt 5110 Oncon0 6159 suc csuc 6161 ‘cfv 6324 setrecscsetrecs 45213 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-reg 9040 ax-inf2 9088 |
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 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-r1 9177 df-rank 9178 df-setrecs 45214 |
This theorem is referenced by: (None) |
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