isfin1-2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > isfin1-2		Structured version Visualization version GIF version

Theorem isfin1-2 10309

Description: A set is finite in the usual sense iff the power set of its power set is Dedekind finite. (Contributed by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
isfin1-2	⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin^IV)

**Proof of Theorem isfin1-2**
Step	Hyp	Ref	Expression
1		elex 3463	. 2 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V)
2		elex 3463	. . 3 ⊢ (𝒫 𝒫 𝐴 ∈ Fin^IV → 𝒫 𝒫 𝐴 ∈ V)
3		pwexb 7723	. . . 4 ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
4		pwexb 7723	. . . 4 ⊢ (𝒫 𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V)
5	3, 4	bitri 275	. . 3 ⊢ (𝐴 ∈ V ↔ 𝒫 𝒫 𝐴 ∈ V)
6	2, 5	sylibr 234	. 2 ⊢ (𝒫 𝒫 𝐴 ∈ Fin^IV → 𝐴 ∈ V)
7		ominf 9178	. . . . . 6 ⊢ ¬ ω ∈ Fin
8		pwfi 9233	. . . . . . . 8 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
9		pwfi 9233	. . . . . . . 8 ⊢ (𝒫 𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin)
10	8, 9	bitri 275	. . . . . . 7 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin)
11		domfi 9127	. . . . . . . 8 ⊢ ((𝒫 𝒫 𝐴 ∈ Fin ∧ ω ≼ 𝒫 𝒫 𝐴) → ω ∈ Fin)
12	11	expcom 413	. . . . . . 7 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → (𝒫 𝒫 𝐴 ∈ Fin → ω ∈ Fin))
13	10, 12	biimtrid 242	. . . . . 6 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → (𝐴 ∈ Fin → ω ∈ Fin))
14	7, 13	mtoi 199	. . . . 5 ⊢ (ω ≼ 𝒫 𝒫 𝐴 → ¬ 𝐴 ∈ Fin)
15		fineqvlem 9180	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)
16	15	ex 412	. . . . 5 ⊢ (𝐴 ∈ V → (¬ 𝐴 ∈ Fin → ω ≼ 𝒫 𝒫 𝐴))
17	14, 16	impbid2 226	. . . 4 ⊢ (𝐴 ∈ V → (ω ≼ 𝒫 𝒫 𝐴 ↔ ¬ 𝐴 ∈ Fin))
18	17	con2bid 354	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin ↔ ¬ ω ≼ 𝒫 𝒫 𝐴))
19		isfin4-2 10238	. . . 4 ⊢ (𝒫 𝒫 𝐴 ∈ V → (𝒫 𝒫 𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝒫 𝒫 𝐴))
20	5, 19	sylbi 217	. . 3 ⊢ (𝐴 ∈ V → (𝒫 𝒫 𝐴 ∈ Fin^IV ↔ ¬ ω ≼ 𝒫 𝒫 𝐴))
21	18, 20	bitr4d 282	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin^IV))
22	1, 6, 21	pm5.21nii 378	1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝒫 𝐴 ∈ Fin^IV)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 ∈ wcel 2114 Vcvv 3442 𝒫 cpw 4556 class class class wbr 5100 ωcom 7820 ≼ cdom 8895 Fincfn 8897 Fin^IVcfin4 10204

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-ov 7373 df-om 7821 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-er 8647 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fin4 10211

This theorem is referenced by: (None)

W3C validator