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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 3185 | . 2 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V) | |
2 | elex 3185 | . . 3 ⊢ (𝒫 𝒫 𝐴 ∈ FinIV → 𝒫 𝒫 𝐴 ∈ V) | |
3 | pwexb 6867 | . . . 4 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V) | |
4 | pwexb 6867 | . . . 4 ⊢ (𝒫 𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V) | |
5 | 3, 4 | bitri 263 | . . 3 ⊢ (𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V) |
6 | 2, 5 | sylibr 223 | . 2 ⊢ (𝒫 𝒫 𝐴 ∈ FinIV → 𝐴 ∈ V) |
7 | ominf 8057 | . . . . . 6 ⊢ ¬ ω ∈ Fin | |
8 | pwfi 8144 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) | |
9 | pwfi 8144 | . . . . . . . 8 ⊢ (𝒫 𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin) | |
10 | 8, 9 | bitri 263 | . . . . . . 7 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin) |
11 | domfi 8066 | . . . . . . . 8 ⊢ ((𝒫 𝒫 𝐴 ∈ Fin ∧ ω ≼ 𝒫 𝒫 𝐴) → ω ∈ Fin) | |
12 | 11 | expcom 450 | . . . . . . 7 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → (𝒫 𝒫 𝐴 ∈ Fin → ω ∈ Fin)) |
13 | 10, 12 | syl5bi 231 | . . . . . 6 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → (𝐴 ∈ Fin → ω ∈ Fin)) |
14 | 7, 13 | mtoi 189 | . . . . 5 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → ¬ 𝐴 ∈ Fin) |
15 | fineqvlem 8059 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴) | |
16 | 15 | ex 449 | . . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ Fin → ω ≼ 𝒫 𝒫 𝐴)) |
17 | 14, 16 | impbid2 215 | . . . 4 ⊢ (𝐴 ∈ V → (ω ≼ 𝒫 𝒫 𝐴 ↔ ¬ 𝐴 ∈ Fin)) |
18 | 17 | con2bid 343 | . . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin ↔ ¬ ω ≼ 𝒫 𝒫 𝐴)) |
19 | isfin4-2 9019 | . . . 4 ⊢ (𝒫 𝒫 𝐴 ∈ V → (𝒫 𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝒫 𝐴)) | |
20 | 5, 19 | sylbi 206 | . . 3 ⊢ (𝐴 ∈ V → (𝒫 𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝒫 𝐴)) |
21 | 18, 20 | bitr4d 270 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ FinIV)) |
22 | 1, 6, 21 | pm5.21nii 367 | 1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ FinIV) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∈ wcel 1977 Vcvv 3173 𝒫 cpw 4108 class class class wbr 4583 ωcom 6957 ≼ cdom 7839 Fincfn 7841 FinIVcfin4 8985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fin4 8992 |
This theorem is referenced by: (None) |
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