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Mirrors > Home > MPE Home > Th. List > pwfi | Structured version Visualization version GIF version |
Description: The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.) |
Ref | Expression |
---|---|
pwfi | ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 8516 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑚 ∈ ω 𝐴 ≈ 𝑚) | |
2 | pweq 4513 | . . . . . . 7 ⊢ (𝑚 = ∅ → 𝒫 𝑚 = 𝒫 ∅) | |
3 | 2 | eleq1d 2874 | . . . . . 6 ⊢ (𝑚 = ∅ → (𝒫 𝑚 ∈ Fin ↔ 𝒫 ∅ ∈ Fin)) |
4 | pweq 4513 | . . . . . . 7 ⊢ (𝑚 = 𝑘 → 𝒫 𝑚 = 𝒫 𝑘) | |
5 | 4 | eleq1d 2874 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 𝑘 ∈ Fin)) |
6 | pweq 4513 | . . . . . . . 8 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 suc 𝑘) | |
7 | df-suc 6165 | . . . . . . . . 9 ⊢ suc 𝑘 = (𝑘 ∪ {𝑘}) | |
8 | 7 | pweqi 4515 | . . . . . . . 8 ⊢ 𝒫 suc 𝑘 = 𝒫 (𝑘 ∪ {𝑘}) |
9 | 6, 8 | eqtrdi 2849 | . . . . . . 7 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 (𝑘 ∪ {𝑘})) |
10 | 9 | eleq1d 2874 | . . . . . 6 ⊢ (𝑚 = suc 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 (𝑘 ∪ {𝑘}) ∈ Fin)) |
11 | pw0 4705 | . . . . . . . 8 ⊢ 𝒫 ∅ = {∅} | |
12 | df1o2 8099 | . . . . . . . 8 ⊢ 1o = {∅} | |
13 | 11, 12 | eqtr4i 2824 | . . . . . . 7 ⊢ 𝒫 ∅ = 1o |
14 | 1onn 8248 | . . . . . . . 8 ⊢ 1o ∈ ω | |
15 | ssid 3937 | . . . . . . . 8 ⊢ 1o ⊆ 1o | |
16 | ssnnfi 8721 | . . . . . . . 8 ⊢ ((1o ∈ ω ∧ 1o ⊆ 1o) → 1o ∈ Fin) | |
17 | 14, 15, 16 | mp2an 691 | . . . . . . 7 ⊢ 1o ∈ Fin |
18 | 13, 17 | eqeltri 2886 | . . . . . 6 ⊢ 𝒫 ∅ ∈ Fin |
19 | eqid 2798 | . . . . . . . 8 ⊢ (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘})) = (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘})) | |
20 | 19 | pwfilem 8802 | . . . . . . 7 ⊢ (𝒫 𝑘 ∈ Fin → 𝒫 (𝑘 ∪ {𝑘}) ∈ Fin) |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ω → (𝒫 𝑘 ∈ Fin → 𝒫 (𝑘 ∪ {𝑘}) ∈ Fin)) |
22 | 3, 5, 10, 18, 21 | finds1 7592 | . . . . 5 ⊢ (𝑚 ∈ ω → 𝒫 𝑚 ∈ Fin) |
23 | pwen 8674 | . . . . 5 ⊢ (𝐴 ≈ 𝑚 → 𝒫 𝐴 ≈ 𝒫 𝑚) | |
24 | enfii 8719 | . . . . 5 ⊢ ((𝒫 𝑚 ∈ Fin ∧ 𝒫 𝐴 ≈ 𝒫 𝑚) → 𝒫 𝐴 ∈ Fin) | |
25 | 22, 23, 24 | syl2an 598 | . . . 4 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ 𝑚) → 𝒫 𝐴 ∈ Fin) |
26 | 25 | rexlimiva 3240 | . . 3 ⊢ (∃𝑚 ∈ ω 𝐴 ≈ 𝑚 → 𝒫 𝐴 ∈ Fin) |
27 | 1, 26 | sylbi 220 | . 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin) |
28 | pwexr 7467 | . . . 4 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ V) | |
29 | canth2g 8655 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴) | |
30 | sdomdom 8520 | . . . 4 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴) | |
31 | 28, 29, 30 | 3syl 18 | . . 3 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ≼ 𝒫 𝐴) |
32 | domfi 8723 | . . 3 ⊢ ((𝒫 𝐴 ∈ Fin ∧ 𝐴 ≼ 𝒫 𝐴) → 𝐴 ∈ Fin) | |
33 | 31, 32 | mpdan 686 | . 2 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ Fin) |
34 | 27, 33 | impbii 212 | 1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 suc csuc 6161 ωcom 7560 1oc1o 8078 ≈ cen 8489 ≼ cdom 8490 ≺ csdm 8491 Fincfn 8492 |
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-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 |
This theorem is referenced by: mapfi 8804 r1fin 9186 dfac12k 9558 pwsdompw 9615 ackbij1lem5 9635 ackbij1lem9 9639 ackbij1lem10 9640 ackbij1lem14 9644 ackbij1b 9650 isfin1-2 9796 isfin1-3 9797 domtriomlem 9853 dominf 9856 dominfac 9984 gchhar 10090 omina 10102 gchina 10110 hashpw 13793 hashbclem 13806 qshash 15174 ackbijnn 15175 incexclem 15183 incexc 15184 incexc2 15185 hashbccl 16329 lagsubg2 18333 lagsubg 18334 orbsta2 18436 sylow1lem3 18717 sylow1lem5 18719 sylow2alem2 18735 sylow2a 18736 sylow2blem2 18738 sylow2blem3 18739 sylow3lem3 18746 sylow3lem4 18747 sylow3lem6 18749 pgpfac1lem5 19194 discmp 22003 cmpfi 22013 dis1stc 22104 1stckgenlem 22158 ptcmpfi 22418 fiufl 22521 musum 25776 qerclwwlknfi 27858 hasheuni 31454 coinfliplem 31846 ballotth 31905 erdszelem2 32552 kelac2lem 40008 pwinfig 40260 |
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