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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 48022 and onsetreclem3 48023 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 35297. 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 2726 | . . . 4 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | onsetrec.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
3 | 2 | onsetreclem2 48022 | . . . . . 6 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) |
4 | 3 | ax-gen 1789 | . . . . 5 ⊢ ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)) |
6 | 1, 5 | setrec2v 48012 | . . 3 ⊢ (⊤ → setrecs(𝐹) ⊆ On) |
7 | 6 | mptru 1540 | . 2 ⊢ setrecs(𝐹) ⊆ On |
8 | vex 3472 | . . . . . . 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 48007 | . . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹)) |
12 | 2 | onsetreclem3 48023 | . . . . 5 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) |
13 | ssel 3970 | . . . . 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 3057 | . 2 ⊢ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) |
17 | tfi 7839 | . 2 ⊢ ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On) | |
18 | 7, 16, 17 | mp2an 689 | 1 ⊢ setrecs(𝐹) = On |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ⊆ wss 3943 {cpr 4625 ∪ cuni 4902 ↦ cmpt 5224 Oncon0 6358 suc csuc 6360 ‘cfv 6537 setrecscsetrecs 47999 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-reg 9589 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-r1 9761 df-rank 9762 df-setrecs 48000 |
This theorem is referenced by: (None) |
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