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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 5288. (Revised by BTernaryTau, 7-Sep-2024.) |
Ref | Expression |
---|---|
pwfi | ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4549 | . . . 4 ⊢ (𝑥 = ∅ → 𝒫 𝑥 = 𝒫 ∅) | |
2 | 1 | eleq1d 2823 | . . 3 ⊢ (𝑥 = ∅ → (𝒫 𝑥 ∈ Fin ↔ 𝒫 ∅ ∈ Fin)) |
3 | pweq 4549 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦) | |
4 | 3 | eleq1d 2823 | . . 3 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)) |
5 | pweq 4549 | . . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → 𝒫 𝑥 = 𝒫 (𝑦 ∪ {𝑧})) | |
6 | 5 | eleq1d 2823 | . . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝒫 𝑥 ∈ Fin ↔ 𝒫 (𝑦 ∪ {𝑧}) ∈ Fin)) |
7 | pweq 4549 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
8 | 7 | eleq1d 2823 | . . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)) |
9 | pw0 4745 | . . . 4 ⊢ 𝒫 ∅ = {∅} | |
10 | snfi 8834 | . . . 4 ⊢ {∅} ∈ Fin | |
11 | 9, 10 | eqeltri 2835 | . . 3 ⊢ 𝒫 ∅ ∈ Fin |
12 | eqid 2738 | . . . . 5 ⊢ (𝑐 ∈ 𝒫 𝑦 ↦ (𝑐 ∪ {𝑧})) = (𝑐 ∈ 𝒫 𝑦 ↦ (𝑐 ∪ {𝑧})) | |
13 | 12 | pwfilem 8960 | . . . 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 8947 | . 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin) |
16 | pwfir 8959 | . 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 1539 ∈ wcel 2106 ∪ cun 3885 ∅c0 4256 𝒫 cpw 4533 {csn 4561 ↦ cmpt 5157 Fincfn 8733 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-om 7713 df-1o 8297 df-en 8734 df-fin 8737 |
This theorem is referenced by: mapfi 9115 r1fin 9531 dfac12k 9903 pwsdompw 9960 ackbij1lem5 9980 ackbij1lem9 9984 ackbij1lem10 9985 ackbij1lem14 9989 ackbij1b 9995 isfin1-2 10141 isfin1-3 10142 domtriomlem 10198 dominf 10201 dominfac 10329 gchhar 10435 omina 10447 gchina 10455 hashpw 14151 hashbclem 14164 qshash 15539 ackbijnn 15540 incexclem 15548 incexc 15549 incexc2 15550 hashbccl 16704 lagsubg2 18817 lagsubg 18818 orbsta2 18920 sylow1lem3 19205 sylow1lem5 19207 sylow2alem2 19223 sylow2a 19224 sylow2blem2 19226 sylow2blem3 19227 sylow3lem3 19234 sylow3lem4 19235 sylow3lem6 19237 pgpfac1lem5 19682 discmp 22549 cmpfi 22559 dis1stc 22650 1stckgenlem 22704 ptcmpfi 22964 fiufl 23067 musum 26340 qerclwwlknfi 28437 hasheuni 32053 coinfliplem 32445 ballotth 32504 fineqvpow 33065 erdszelem2 33154 sticksstones22 40124 kelac2lem 40889 pwinfig 41168 |
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