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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 3501 | . 2 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V) | |
| 2 | elex 3501 | . . 3 ⊢ (𝒫 𝒫 𝐴 ∈ FinIV → 𝒫 𝒫 𝐴 ∈ V) | |
| 3 | pwexb 7786 | . . . 4 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | |
| 4 | pwexb 7786 | . . . 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 9294 | . . . . . 6 ⊢ ¬ ω ∈ Fin | |
| 8 | pwfi 9357 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) | |
| 9 | pwfi 9357 | . . . . . . . 8 ⊢ (𝒫 𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin) | |
| 10 | 8, 9 | bitri 275 | . . . . . . 7 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin) |
| 11 | domfi 9229 | . . . . . . . 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 9298 | . . . . . 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 10354 | . . . 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 2108 Vcvv 3480 𝒫 cpw 4600 class class class wbr 5143 ωcom 7887 ≼ cdom 8983 Fincfn 8985 FinIVcfin4 10320 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fin4 10327 |
| This theorem is referenced by: (None) |
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