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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 3498 | . 2 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V) | |
2 | elex 3498 | . . 3 ⊢ (𝒫 𝒫 𝐴 ∈ FinIV → 𝒫 𝒫 𝐴 ∈ V) | |
3 | pwexb 7784 | . . . 4 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | |
4 | pwexb 7784 | . . . 4 ⊢ (𝒫 𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V) | |
5 | 3, 4 | bitri 275 | . . 3 ⊢ (𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V) |
6 | 2, 5 | sylibr 234 | . 2 ⊢ (𝒫 𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V) |
7 | ominf 9291 | . . . . . 6 ⊢ ¬ ω ∈ Fin | |
8 | pwfi 9354 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) | |
9 | pwfi 9354 | . . . . . . . 8 ⊢ (𝒫 𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin) | |
10 | 8, 9 | bitri 275 | . . . . . . 7 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin) |
11 | domfi 9226 | . . . . . . . 8 ⊢ ((𝒫 𝒫 𝐴 ∈ Fin ∧ ω ≼ 𝒫 𝒫 𝐴) → ω ∈ Fin) | |
12 | 11 | expcom 413 | . . . . . . 7 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → (𝒫 𝒫 𝐴 ∈ Fin → ω ∈ Fin)) |
13 | 10, 12 | biimtrid 242 | . . . . . 6 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → (𝐴 ∈ Fin → ω ∈ Fin)) |
14 | 7, 13 | mtoi 199 | . . . . 5 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → ¬ 𝐴 ∈ Fin) |
15 | fineqvlem 9295 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴) | |
16 | 15 | ex 412 | . . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ Fin → ω ≼ 𝒫 𝒫 𝐴)) |
17 | 14, 16 | impbid2 226 | . . . 4 ⊢ (𝐴 ∈ V → (ω ≼ 𝒫 𝒫 𝐴 ↔ ¬ 𝐴 ∈ Fin)) |
18 | 17 | con2bid 354 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin ↔ ¬ ω ≼ 𝒫 𝒫 𝐴)) |
19 | isfin4-2 10351 | . . . 4 ⊢ (𝒫 𝒫 𝐴 ∈ V → (𝒫 𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝒫 𝐴)) | |
20 | 5, 19 | sylbi 217 | . . 3 ⊢ (𝐴 ∈ V → (𝒫 𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝒫 𝐴)) |
21 | 18, 20 | bitr4d 282 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ FinIV)) |
22 | 1, 6, 21 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∈ wcel 2105 Vcvv 3477 𝒫 cpw 4604 class class class wbr 5147 ωcom 7886 ≼ cdom 8981 Fincfn 8983 FinIVcfin4 10317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fin4 10324 |
This theorem is referenced by: (None) |
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