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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 6670 | . . 3 ⊢ (𝑛 = ∅ → (𝑅1‘𝑛) = (𝑅1‘∅)) | |
2 | 1 | eleq1d 2897 | . 2 ⊢ (𝑛 = ∅ → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘∅) ∈ Fin)) |
3 | fveq2 6670 | . . 3 ⊢ (𝑛 = 𝑚 → (𝑅1‘𝑛) = (𝑅1‘𝑚)) | |
4 | 3 | eleq1d 2897 | . 2 ⊢ (𝑛 = 𝑚 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘𝑚) ∈ Fin)) |
5 | fveq2 6670 | . . 3 ⊢ (𝑛 = suc 𝑚 → (𝑅1‘𝑛) = (𝑅1‘suc 𝑚)) | |
6 | 5 | eleq1d 2897 | . 2 ⊢ (𝑛 = suc 𝑚 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘suc 𝑚) ∈ Fin)) |
7 | fveq2 6670 | . . 3 ⊢ (𝑛 = 𝐴 → (𝑅1‘𝑛) = (𝑅1‘𝐴)) | |
8 | 7 | eleq1d 2897 | . 2 ⊢ (𝑛 = 𝐴 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘𝐴) ∈ Fin)) |
9 | r10 9197 | . . 3 ⊢ (𝑅1‘∅) = ∅ | |
10 | 0fin 8746 | . . 3 ⊢ ∅ ∈ Fin | |
11 | 9, 10 | eqeltri 2909 | . 2 ⊢ (𝑅1‘∅) ∈ Fin |
12 | r1funlim 9195 | . . . . . . . . 9 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
13 | 12 | simpri 488 | . . . . . . . 8 ⊢ Lim dom 𝑅1 |
14 | limomss 7585 | . . . . . . . 8 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ ω ⊆ dom 𝑅1 |
16 | 15 | sseli 3963 | . . . . . 6 ⊢ (𝑚 ∈ ω → 𝑚 ∈ dom 𝑅1) |
17 | r1sucg 9198 | . . . . . 6 ⊢ (𝑚 ∈ dom 𝑅1 → (𝑅1‘suc 𝑚) = 𝒫 (𝑅1‘𝑚)) | |
18 | 16, 17 | syl 17 | . . . . 5 ⊢ (𝑚 ∈ ω → (𝑅1‘suc 𝑚) = 𝒫 (𝑅1‘𝑚)) |
19 | 18 | eleq1d 2897 | . . . 4 ⊢ (𝑚 ∈ ω → ((𝑅1‘suc 𝑚) ∈ Fin ↔ 𝒫 (𝑅1‘𝑚) ∈ Fin)) |
20 | pwfi 8819 | . . . 4 ⊢ ((𝑅1‘𝑚) ∈ Fin ↔ 𝒫 (𝑅1‘𝑚) ∈ Fin) | |
21 | 19, 20 | syl6rbbr 292 | . . 3 ⊢ (𝑚 ∈ ω → ((𝑅1‘𝑚) ∈ Fin ↔ (𝑅1‘suc 𝑚) ∈ Fin)) |
22 | 21 | biimpd 231 | . 2 ⊢ (𝑚 ∈ ω → ((𝑅1‘𝑚) ∈ Fin → (𝑅1‘suc 𝑚) ∈ Fin)) |
23 | 2, 4, 6, 8, 11, 22 | finds 7608 | 1 ⊢ (𝐴 ∈ ω → (𝑅1‘𝐴) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 dom cdm 5555 Lim wlim 6192 suc csuc 6193 Fun wfun 6349 ‘cfv 6355 ωcom 7580 Fincfn 8509 𝑅1cr1 9191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-r1 9193 |
This theorem is referenced by: ackbij2lem2 9662 ackbij2 9665 |
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