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Theorem isfin1-4 10427
Description: A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
isfin1-4 (𝐴𝑉 → (𝐴 ∈ Fin ↔ [] Fr 𝒫 𝐴))

Proof of Theorem isfin1-4
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isfin1-3 10426 . 2 (𝐴𝑉 → (𝐴 ∈ Fin ↔ [] Fr 𝒫 𝐴))
2 eqid 2737 . . . 4 (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥))
32compssiso 10414 . . 3 (𝐴𝑉 → (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) Isom [] , [] (𝒫 𝐴, 𝒫 𝐴))
4 isofr 7362 . . 3 ((𝑥 ∈ 𝒫 𝐴 ↦ (𝐴𝑥)) Isom [] , [] (𝒫 𝐴, 𝒫 𝐴) → ( [] Fr 𝒫 𝐴 [] Fr 𝒫 𝐴))
53, 4syl 17 . 2 (𝐴𝑉 → ( [] Fr 𝒫 𝐴 [] Fr 𝒫 𝐴))
61, 5bitr4d 282 1 (𝐴𝑉 → (𝐴 ∈ Fin ↔ [] Fr 𝒫 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  cdif 3948  𝒫 cpw 4600  cmpt 5225   Fr wfr 5634  ccnv 5684   Isom wiso 6562   [] crpss 7742  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-rpss 7743  df-om 7888  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989
This theorem is referenced by: (None)
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