![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isfin1-2 | Structured version Visualization version GIF version |
Description: A set is finite in the usual sense iff the power set of its power set is Dedekind finite. (Contributed by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
isfin1-2 | ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ FinIV) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3459 | . 2 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V) | |
2 | elex 3459 | . . 3 ⊢ (𝒫 𝒫 𝐴 ∈ FinIV → 𝒫 𝒫 𝐴 ∈ V) | |
3 | pwexb 7468 | . . . 4 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | |
4 | pwexb 7468 | . . . 4 ⊢ (𝒫 𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V) | |
5 | 3, 4 | bitri 278 | . . 3 ⊢ (𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V) |
6 | 2, 5 | sylibr 237 | . 2 ⊢ (𝒫 𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V) |
7 | ominf 8714 | . . . . . 6 ⊢ ¬ ω ∈ Fin | |
8 | pwfi 8803 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) | |
9 | pwfi 8803 | . . . . . . . 8 ⊢ (𝒫 𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin) | |
10 | 8, 9 | bitri 278 | . . . . . . 7 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin) |
11 | domfi 8723 | . . . . . . . 8 ⊢ ((𝒫 𝒫 𝐴 ∈ Fin ∧ ω ≼ 𝒫 𝒫 𝐴) → ω ∈ Fin) | |
12 | 11 | expcom 417 | . . . . . . 7 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → (𝒫 𝒫 𝐴 ∈ Fin → ω ∈ Fin)) |
13 | 10, 12 | syl5bi 245 | . . . . . 6 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → (𝐴 ∈ Fin → ω ∈ Fin)) |
14 | 7, 13 | mtoi 202 | . . . . 5 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → ¬ 𝐴 ∈ Fin) |
15 | fineqvlem 8716 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴) | |
16 | 15 | ex 416 | . . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ Fin → ω ≼ 𝒫 𝒫 𝐴)) |
17 | 14, 16 | impbid2 229 | . . . 4 ⊢ (𝐴 ∈ V → (ω ≼ 𝒫 𝒫 𝐴 ↔ ¬ 𝐴 ∈ Fin)) |
18 | 17 | con2bid 358 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin ↔ ¬ ω ≼ 𝒫 𝒫 𝐴)) |
19 | isfin4-2 9725 | . . . 4 ⊢ (𝒫 𝒫 𝐴 ∈ V → (𝒫 𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝒫 𝐴)) | |
20 | 5, 19 | sylbi 220 | . . 3 ⊢ (𝐴 ∈ V → (𝒫 𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝒫 𝐴)) |
21 | 18, 20 | bitr4d 285 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ FinIV)) |
22 | 1, 6, 21 | pm5.21nii 383 | 1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∈ wcel 2111 Vcvv 3441 𝒫 cpw 4497 class class class wbr 5030 ωcom 7560 ≼ cdom 8490 Fincfn 8492 FinIVcfin4 9691 |
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-rep 5154 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-fin4 9698 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |