![]() |
Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
|
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 48937 and onsetreclem3 48938 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 35774. 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 2735 | . . . 4 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
2 | onsetrec.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
3 | 2 | onsetreclem2 48937 | . . . . . 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 48927 | . . 3 ⊢ (⊤ → setrecs(𝐹) ⊆ On) |
7 | 6 | mptru 1544 | . 2 ⊢ setrecs(𝐹) ⊆ On |
8 | vex 3482 | . . . . . . 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 48922 | . . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹)) |
12 | 2 | onsetreclem3 48938 | . . . . 5 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎)) |
13 | ssel 3989 | . . . . 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 3061 | . 2 ⊢ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) |
17 | tfi 7874 | . 2 ⊢ ((setrecs(𝐹) ⊆ On ∧ ∀𝑎 ∈ On (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹))) → setrecs(𝐹) = On) | |
18 | 7, 16, 17 | mp2an 692 | 1 ⊢ setrecs(𝐹) = On |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⊆ wss 3963 {cpr 4633 ∪ cuni 4912 ↦ cmpt 5231 Oncon0 6386 suc csuc 6388 ‘cfv 6563 setrecscsetrecs 48914 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-reg 9630 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-r1 9802 df-rank 9803 df-setrecs 48915 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |