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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.) Avoid ax-pow 5362. (Revised by BTernaryTau, 7-Sep-2024.) |
Ref | Expression |
---|---|
pwfi | ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4615 | . . . 4 ⊢ (𝑥 = ∅ → 𝒫 𝑥 = 𝒫 ∅) | |
2 | 1 | eleq1d 2819 | . . 3 ⊢ (𝑥 = ∅ → (𝒫 𝑥 ∈ Fin ↔ 𝒫 ∅ ∈ Fin)) |
3 | pweq 4615 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦) | |
4 | 3 | eleq1d 2819 | . . 3 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)) |
5 | pweq 4615 | . . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → 𝒫 𝑥 = 𝒫 (𝑦 ∪ {𝑧})) | |
6 | 5 | eleq1d 2819 | . . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝒫 𝑥 ∈ Fin ↔ 𝒫 (𝑦 ∪ {𝑧}) ∈ Fin)) |
7 | pweq 4615 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
8 | 7 | eleq1d 2819 | . . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)) |
9 | pw0 4814 | . . . 4 ⊢ 𝒫 ∅ = {∅} | |
10 | snfi 9040 | . . . 4 ⊢ {∅} ∈ Fin | |
11 | 9, 10 | eqeltri 2830 | . . 3 ⊢ 𝒫 ∅ ∈ Fin |
12 | eqid 2733 | . . . . 5 ⊢ (𝑐 ∈ 𝒫 𝑦 ↦ (𝑐 ∪ {𝑧})) = (𝑐 ∈ 𝒫 𝑦 ↦ (𝑐 ∪ {𝑧})) | |
13 | 12 | pwfilem 9173 | . . . 4 ⊢ (𝒫 𝑦 ∈ Fin → 𝒫 (𝑦 ∪ {𝑧}) ∈ Fin) |
14 | 13 | a1i 11 | . . 3 ⊢ (𝑦 ∈ Fin → (𝒫 𝑦 ∈ Fin → 𝒫 (𝑦 ∪ {𝑧}) ∈ Fin)) |
15 | 2, 4, 6, 8, 11, 14 | findcard2 9160 | . 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin) |
16 | pwfir 9172 | . 2 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ Fin) | |
17 | 15, 16 | impbii 208 | 1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∪ cun 3945 ∅c0 4321 𝒫 cpw 4601 {csn 4627 ↦ cmpt 5230 Fincfn 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7851 df-1o 8461 df-en 8936 df-fin 8939 |
This theorem is referenced by: xpfi 9313 mapfi 9344 r1fin 9764 dfac12k 10138 pwsdompw 10195 ackbij1lem5 10215 ackbij1lem9 10219 ackbij1lem10 10220 ackbij1lem14 10224 ackbij1b 10230 isfin1-2 10376 isfin1-3 10377 domtriomlem 10433 dominf 10436 dominfac 10564 gchhar 10670 omina 10682 gchina 10690 hashpw 14392 hashbclem 14407 qshash 15769 ackbijnn 15770 incexclem 15778 incexc 15779 incexc2 15780 hashbccl 16932 lagsubg2 19065 lagsubg 19066 orbsta2 19172 sylow1lem3 19461 sylow1lem5 19463 sylow2alem2 19479 sylow2a 19480 sylow2blem2 19482 sylow2blem3 19483 sylow3lem3 19490 sylow3lem4 19491 sylow3lem6 19493 pgpfac1lem5 19941 discmp 22884 cmpfi 22894 dis1stc 22985 1stckgenlem 23039 ptcmpfi 23299 fiufl 23402 musum 26675 qerclwwlknfi 29306 hasheuni 33021 coinfliplem 33415 ballotth 33474 fineqvpow 34034 erdszelem2 34121 sticksstones22 40922 kelac2lem 41739 pwinfig 42245 |
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