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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 8533 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑚 ∈ ω 𝐴 ≈ 𝑚) | |
2 | pweq 4555 | . . . . . . 7 ⊢ (𝑚 = ∅ → 𝒫 𝑚 = 𝒫 ∅) | |
3 | 2 | eleq1d 2897 | . . . . . 6 ⊢ (𝑚 = ∅ → (𝒫 𝑚 ∈ Fin ↔ 𝒫 ∅ ∈ Fin)) |
4 | pweq 4555 | . . . . . . 7 ⊢ (𝑚 = 𝑘 → 𝒫 𝑚 = 𝒫 𝑘) | |
5 | 4 | eleq1d 2897 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 𝑘 ∈ Fin)) |
6 | pweq 4555 | . . . . . . . 8 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 suc 𝑘) | |
7 | df-suc 6197 | . . . . . . . . 9 ⊢ suc 𝑘 = (𝑘 ∪ {𝑘}) | |
8 | 7 | pweqi 4557 | . . . . . . . 8 ⊢ 𝒫 suc 𝑘 = 𝒫 (𝑘 ∪ {𝑘}) |
9 | 6, 8 | syl6eq 2872 | . . . . . . 7 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 (𝑘 ∪ {𝑘})) |
10 | 9 | eleq1d 2897 | . . . . . 6 ⊢ (𝑚 = suc 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 (𝑘 ∪ {𝑘}) ∈ Fin)) |
11 | pw0 4745 | . . . . . . . 8 ⊢ 𝒫 ∅ = {∅} | |
12 | df1o2 8116 | . . . . . . . 8 ⊢ 1o = {∅} | |
13 | 11, 12 | eqtr4i 2847 | . . . . . . 7 ⊢ 𝒫 ∅ = 1o |
14 | 1onn 8265 | . . . . . . . 8 ⊢ 1o ∈ ω | |
15 | ssid 3989 | . . . . . . . 8 ⊢ 1o ⊆ 1o | |
16 | ssnnfi 8737 | . . . . . . . 8 ⊢ ((1o ∈ ω ∧ 1o ⊆ 1o) → 1o ∈ Fin) | |
17 | 14, 15, 16 | mp2an 690 | . . . . . . 7 ⊢ 1o ∈ Fin |
18 | 13, 17 | eqeltri 2909 | . . . . . 6 ⊢ 𝒫 ∅ ∈ Fin |
19 | eqid 2821 | . . . . . . . 8 ⊢ (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘})) = (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘})) | |
20 | 19 | pwfilem 8818 | . . . . . . 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 7611 | . . . . 5 ⊢ (𝑚 ∈ ω → 𝒫 𝑚 ∈ Fin) |
23 | pwen 8690 | . . . . 5 ⊢ (𝐴 ≈ 𝑚 → 𝒫 𝐴 ≈ 𝒫 𝑚) | |
24 | enfii 8735 | . . . . 5 ⊢ ((𝒫 𝑚 ∈ Fin ∧ 𝒫 𝐴 ≈ 𝒫 𝑚) → 𝒫 𝐴 ∈ Fin) | |
25 | 22, 23, 24 | syl2an 597 | . . . 4 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ 𝑚) → 𝒫 𝐴 ∈ Fin) |
26 | 25 | rexlimiva 3281 | . . 3 ⊢ (∃𝑚 ∈ ω 𝐴 ≈ 𝑚 → 𝒫 𝐴 ∈ Fin) |
27 | 1, 26 | sylbi 219 | . 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin) |
28 | pwexr 7487 | . . . 4 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ V) | |
29 | canth2g 8671 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴) | |
30 | sdomdom 8537 | . . . 4 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴) | |
31 | 28, 29, 30 | 3syl 18 | . . 3 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ≼ 𝒫 𝐴) |
32 | domfi 8739 | . . 3 ⊢ ((𝒫 𝐴 ∈ Fin ∧ 𝐴 ≼ 𝒫 𝐴) → 𝐴 ∈ Fin) | |
33 | 31, 32 | mpdan 685 | . 2 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ Fin) |
34 | 27, 33 | impbii 211 | 1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 ∪ cun 3934 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 {csn 4567 class class class wbr 5066 ↦ cmpt 5146 suc csuc 6193 ωcom 7580 1oc1o 8095 ≈ cen 8506 ≼ cdom 8507 ≺ csdm 8508 Fincfn 8509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 |
This theorem is referenced by: mapfi 8820 r1fin 9202 dfac12k 9573 pwsdompw 9626 ackbij1lem5 9646 ackbij1lem9 9650 ackbij1lem10 9651 ackbij1lem14 9655 ackbij1b 9661 isfin1-2 9807 isfin1-3 9808 domtriomlem 9864 dominf 9867 dominfac 9995 gchhar 10101 omina 10113 gchina 10121 hashpw 13798 hashbclem 13811 qshash 15182 ackbijnn 15183 incexclem 15191 incexc 15192 incexc2 15193 hashbccl 16339 lagsubg2 18341 lagsubg 18342 orbsta2 18444 sylow1lem3 18725 sylow1lem5 18727 sylow2alem2 18743 sylow2a 18744 sylow2blem2 18746 sylow2blem3 18747 sylow3lem3 18754 sylow3lem4 18755 sylow3lem6 18757 pgpfac1lem5 19201 discmp 22006 cmpfi 22016 dis1stc 22107 1stckgenlem 22161 ptcmpfi 22421 fiufl 22524 musum 25768 qerclwwlknfi 27852 hasheuni 31344 coinfliplem 31736 ballotth 31795 erdszelem2 32439 kelac2lem 39684 pwinfig 39940 |
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