Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > r1fin | Structured version Visualization version GIF version |
Description: The first ω levels of the cumulative hierarchy are all finite. (Contributed by Mario Carneiro, 15-May-2013.) |
Ref | Expression |
---|---|
r1fin | ⊢ (𝐴 ∈ ω → (𝑅1‘𝐴) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6726 | . . 3 ⊢ (𝑛 = ∅ → (𝑅1‘𝑛) = (𝑅1‘∅)) | |
2 | 1 | eleq1d 2823 | . 2 ⊢ (𝑛 = ∅ → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘∅) ∈ Fin)) |
3 | fveq2 6726 | . . 3 ⊢ (𝑛 = 𝑚 → (𝑅1‘𝑛) = (𝑅1‘𝑚)) | |
4 | 3 | eleq1d 2823 | . 2 ⊢ (𝑛 = 𝑚 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘𝑚) ∈ Fin)) |
5 | fveq2 6726 | . . 3 ⊢ (𝑛 = suc 𝑚 → (𝑅1‘𝑛) = (𝑅1‘suc 𝑚)) | |
6 | 5 | eleq1d 2823 | . 2 ⊢ (𝑛 = suc 𝑚 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘suc 𝑚) ∈ Fin)) |
7 | fveq2 6726 | . . 3 ⊢ (𝑛 = 𝐴 → (𝑅1‘𝑛) = (𝑅1‘𝐴)) | |
8 | 7 | eleq1d 2823 | . 2 ⊢ (𝑛 = 𝐴 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘𝐴) ∈ Fin)) |
9 | r10 9397 | . . 3 ⊢ (𝑅1‘∅) = ∅ | |
10 | 0fin 8860 | . . 3 ⊢ ∅ ∈ Fin | |
11 | 9, 10 | eqeltri 2835 | . 2 ⊢ (𝑅1‘∅) ∈ Fin |
12 | pwfi 8867 | . . . 4 ⊢ ((𝑅1‘𝑚) ∈ Fin ↔ 𝒫 (𝑅1‘𝑚) ∈ Fin) | |
13 | r1funlim 9395 | . . . . . . . . 9 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
14 | 13 | simpri 489 | . . . . . . . 8 ⊢ Lim dom 𝑅1 |
15 | limomss 7658 | . . . . . . . 8 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
16 | 14, 15 | ax-mp 5 | . . . . . . 7 ⊢ ω ⊆ dom 𝑅1 |
17 | 16 | sseli 3905 | . . . . . 6 ⊢ (𝑚 ∈ ω → 𝑚 ∈ dom 𝑅1) |
18 | r1sucg 9398 | . . . . . 6 ⊢ (𝑚 ∈ dom 𝑅1 → (𝑅1‘suc 𝑚) = 𝒫 (𝑅1‘𝑚)) | |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝑚 ∈ ω → (𝑅1‘suc 𝑚) = 𝒫 (𝑅1‘𝑚)) |
20 | 19 | eleq1d 2823 | . . . 4 ⊢ (𝑚 ∈ ω → ((𝑅1‘suc 𝑚) ∈ Fin ↔ 𝒫 (𝑅1‘𝑚) ∈ Fin)) |
21 | 12, 20 | bitr4id 293 | . . 3 ⊢ (𝑚 ∈ ω → ((𝑅1‘𝑚) ∈ Fin ↔ (𝑅1‘suc 𝑚) ∈ Fin)) |
22 | 21 | biimpd 232 | . 2 ⊢ (𝑚 ∈ ω → ((𝑅1‘𝑚) ∈ Fin → (𝑅1‘suc 𝑚) ∈ Fin)) |
23 | 2, 4, 6, 8, 11, 22 | finds 7685 | 1 ⊢ (𝐴 ∈ ω → (𝑅1‘𝐴) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2111 ⊆ wss 3875 ∅c0 4246 𝒫 cpw 4522 dom cdm 5560 Lim wlim 6223 suc csuc 6224 Fun wfun 6383 ‘cfv 6389 ωcom 7653 Fincfn 8635 𝑅1cr1 9391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pow 5267 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-om 7654 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-en 8636 df-fin 8639 df-r1 9393 |
This theorem is referenced by: ackbij2lem2 9867 ackbij2 9870 |
Copyright terms: Public domain | W3C validator |