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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 5325. (Revised by BTernaryTau, 7-Sep-2024.) |
Ref | Expression |
---|---|
pwfi | ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pweq 4579 | . . . 4 ⊢ (𝑥 = ∅ → 𝒫 𝑥 = 𝒫 ∅) | |
2 | 1 | eleq1d 2823 | . . 3 ⊢ (𝑥 = ∅ → (𝒫 𝑥 ∈ Fin ↔ 𝒫 ∅ ∈ Fin)) |
3 | pweq 4579 | . . . 4 ⊢ (𝑥 = 𝑦 → 𝒫 𝑥 = 𝒫 𝑦) | |
4 | 3 | eleq1d 2823 | . . 3 ⊢ (𝑥 = 𝑦 → (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)) |
5 | pweq 4579 | . . . 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 4579 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴) | |
8 | 7 | eleq1d 2823 | . . 3 ⊢ (𝑥 = 𝐴 → (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)) |
9 | pw0 4777 | . . . 4 ⊢ 𝒫 ∅ = {∅} | |
10 | snfi 8995 | . . . 4 ⊢ {∅} ∈ Fin | |
11 | 9, 10 | eqeltri 2834 | . . 3 ⊢ 𝒫 ∅ ∈ Fin |
12 | eqid 2737 | . . . . 5 ⊢ (𝑐 ∈ 𝒫 𝑦 ↦ (𝑐 ∪ {𝑧})) = (𝑐 ∈ 𝒫 𝑦 ↦ (𝑐 ∪ {𝑧})) | |
13 | 12 | pwfilem 9128 | . . . 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 9115 | . 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin) |
16 | pwfir 9127 | . 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 3913 ∅c0 4287 𝒫 cpw 4565 {csn 4591 ↦ cmpt 5193 Fincfn 8890 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-om 7808 df-1o 8417 df-en 8891 df-fin 8894 |
This theorem is referenced by: xpfi 9268 mapfi 9299 r1fin 9716 dfac12k 10090 pwsdompw 10147 ackbij1lem5 10167 ackbij1lem9 10171 ackbij1lem10 10172 ackbij1lem14 10176 ackbij1b 10182 isfin1-2 10328 isfin1-3 10329 domtriomlem 10385 dominf 10388 dominfac 10516 gchhar 10622 omina 10634 gchina 10642 hashpw 14343 hashbclem 14356 qshash 15719 ackbijnn 15720 incexclem 15728 incexc 15729 incexc2 15730 hashbccl 16882 lagsubg2 18998 lagsubg 18999 orbsta2 19101 sylow1lem3 19389 sylow1lem5 19391 sylow2alem2 19407 sylow2a 19408 sylow2blem2 19410 sylow2blem3 19411 sylow3lem3 19418 sylow3lem4 19419 sylow3lem6 19421 pgpfac1lem5 19865 discmp 22765 cmpfi 22775 dis1stc 22866 1stckgenlem 22920 ptcmpfi 23180 fiufl 23283 musum 26556 qerclwwlknfi 29059 hasheuni 32724 coinfliplem 33118 ballotth 33177 fineqvpow 33737 erdszelem2 33826 sticksstones22 40605 kelac2lem 41420 pwinfig 41907 |
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