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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 6645 | . . 3 ⊢ (𝑛 = ∅ → (𝑅1‘𝑛) = (𝑅1‘∅)) | |
2 | 1 | eleq1d 2874 | . 2 ⊢ (𝑛 = ∅ → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘∅) ∈ Fin)) |
3 | fveq2 6645 | . . 3 ⊢ (𝑛 = 𝑚 → (𝑅1‘𝑛) = (𝑅1‘𝑚)) | |
4 | 3 | eleq1d 2874 | . 2 ⊢ (𝑛 = 𝑚 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘𝑚) ∈ Fin)) |
5 | fveq2 6645 | . . 3 ⊢ (𝑛 = suc 𝑚 → (𝑅1‘𝑛) = (𝑅1‘suc 𝑚)) | |
6 | 5 | eleq1d 2874 | . 2 ⊢ (𝑛 = suc 𝑚 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘suc 𝑚) ∈ Fin)) |
7 | fveq2 6645 | . . 3 ⊢ (𝑛 = 𝐴 → (𝑅1‘𝑛) = (𝑅1‘𝐴)) | |
8 | 7 | eleq1d 2874 | . 2 ⊢ (𝑛 = 𝐴 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘𝐴) ∈ Fin)) |
9 | r10 9181 | . . 3 ⊢ (𝑅1‘∅) = ∅ | |
10 | 0fin 8730 | . . 3 ⊢ ∅ ∈ Fin | |
11 | 9, 10 | eqeltri 2886 | . 2 ⊢ (𝑅1‘∅) ∈ Fin |
12 | pwfi 8803 | . . . 4 ⊢ ((𝑅1‘𝑚) ∈ Fin ↔ 𝒫 (𝑅1‘𝑚) ∈ Fin) | |
13 | r1funlim 9179 | . . . . . . . . 9 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
14 | 13 | simpri 489 | . . . . . . . 8 ⊢ Lim dom 𝑅1 |
15 | limomss 7565 | . . . . . . . 8 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
16 | 14, 15 | ax-mp 5 | . . . . . . 7 ⊢ ω ⊆ dom 𝑅1 |
17 | 16 | sseli 3911 | . . . . . 6 ⊢ (𝑚 ∈ ω → 𝑚 ∈ dom 𝑅1) |
18 | r1sucg 9182 | . . . . . 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 2874 | . . . 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 7589 | 1 ⊢ (𝐴 ∈ ω → (𝑅1‘𝐴) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 dom cdm 5519 Lim wlim 6160 suc csuc 6161 Fun wfun 6318 ‘cfv 6324 ωcom 7560 Fincfn 8492 𝑅1cr1 9175 |
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-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
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-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-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-r1 9177 |
This theorem is referenced by: ackbij2lem2 9651 ackbij2 9654 |
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