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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 6672 | . . 3 ⊢ (𝑛 = ∅ → (𝑅1‘𝑛) = (𝑅1‘∅)) | |
2 | 1 | eleq1d 2899 | . 2 ⊢ (𝑛 = ∅ → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘∅) ∈ Fin)) |
3 | fveq2 6672 | . . 3 ⊢ (𝑛 = 𝑚 → (𝑅1‘𝑛) = (𝑅1‘𝑚)) | |
4 | 3 | eleq1d 2899 | . 2 ⊢ (𝑛 = 𝑚 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘𝑚) ∈ Fin)) |
5 | fveq2 6672 | . . 3 ⊢ (𝑛 = suc 𝑚 → (𝑅1‘𝑛) = (𝑅1‘suc 𝑚)) | |
6 | 5 | eleq1d 2899 | . 2 ⊢ (𝑛 = suc 𝑚 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘suc 𝑚) ∈ Fin)) |
7 | fveq2 6672 | . . 3 ⊢ (𝑛 = 𝐴 → (𝑅1‘𝑛) = (𝑅1‘𝐴)) | |
8 | 7 | eleq1d 2899 | . 2 ⊢ (𝑛 = 𝐴 → ((𝑅1‘𝑛) ∈ Fin ↔ (𝑅1‘𝐴) ∈ Fin)) |
9 | r10 9199 | . . 3 ⊢ (𝑅1‘∅) = ∅ | |
10 | 0fin 8748 | . . 3 ⊢ ∅ ∈ Fin | |
11 | 9, 10 | eqeltri 2911 | . 2 ⊢ (𝑅1‘∅) ∈ Fin |
12 | r1funlim 9197 | . . . . . . . . 9 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
13 | 12 | simpri 488 | . . . . . . . 8 ⊢ Lim dom 𝑅1 |
14 | limomss 7587 | . . . . . . . 8 ⊢ (Lim dom 𝑅1 → ω ⊆ dom 𝑅1) | |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ ω ⊆ dom 𝑅1 |
16 | 15 | sseli 3965 | . . . . . 6 ⊢ (𝑚 ∈ ω → 𝑚 ∈ dom 𝑅1) |
17 | r1sucg 9200 | . . . . . 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 2899 | . . . 4 ⊢ (𝑚 ∈ ω → ((𝑅1‘suc 𝑚) ∈ Fin ↔ 𝒫 (𝑅1‘𝑚) ∈ Fin)) |
20 | pwfi 8821 | . . . 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 7610 | 1 ⊢ (𝐴 ∈ ω → (𝑅1‘𝐴) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 dom cdm 5557 Lim wlim 6194 suc csuc 6195 Fun wfun 6351 ‘cfv 6357 ωcom 7582 Fincfn 8511 𝑅1cr1 9193 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-r1 9195 |
This theorem is referenced by: ackbij2lem2 9664 ackbij2 9667 |
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