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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 44802 and onsetreclem3 44803 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 33032. 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 2821 | . . . 4 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | onsetrec.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
3 | 2 | onsetreclem2 44802 | . . . . . 6 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) |
4 | 3 | ax-gen 1792 | . . . . 5 ⊢ ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → ∀𝑎(𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)) |
6 | 1, 5 | setrec2v 44793 | . . 3 ⊢ (⊤ → setrecs(𝐹) ⊆ On) |
7 | 6 | mptru 1540 | . 2 ⊢ setrecs(𝐹) ⊆ On |
8 | vex 3497 | . . . . . . 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 44788 | . . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹)) |
12 | 2 | onsetreclem3 44803 | . . . . 5 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) |
13 | ssel 3960 | . . . . 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 3148 | . 2 ⊢ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) |
17 | tfi 7562 | . 2 ⊢ ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On) | |
18 | 7, 16, 17 | mp2an 690 | 1 ⊢ setrecs(𝐹) = On |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 {cpr 4562 ∪ cuni 4831 ↦ cmpt 5138 Oncon0 6185 suc csuc 6187 ‘cfv 6349 setrecscsetrecs 44780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-reg 9050 ax-inf2 9098 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-r1 9187 df-rank 9188 df-setrecs 44781 |
This theorem is referenced by: (None) |
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