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Mirrors > Home > ILE Home > Th. List > enfii | GIF version |
Description: A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) |
Ref | Expression |
---|---|
enfii | ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enfi 6830 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
2 | 1 | biimparc 297 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 class class class wbr 3976 ≈ cen 6695 Fincfn 6697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-er 6492 df-en 6698 df-fin 6700 |
This theorem is referenced by: dif1en 6836 diffisn 6850 xpfi 6886 fisseneq 6888 fundmfi 6894 relcnvfi 6897 f1ofi 6899 f1dmvrnfibi 6900 f1finf1o 6903 en1eqsn 6904 exmidonfinlem 7140 fzfig 10355 hashennnuni 10681 hashennn 10682 summodclem2 11309 zsumdc 11311 prodmodclem2 11504 zproddc 11506 |
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