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Mirrors > Home > MPE Home > Th. List > enfii | Structured version Visualization version 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 8734 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
2 | 1 | biimparc 482 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 ≈ cen 8506 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-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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 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-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-er 8289 df-en 8510 df-fin 8513 |
This theorem is referenced by: domfi 8739 en1eqsn 8748 isfinite2 8776 xpfi 8789 fofinf1o 8799 cnvfi 8806 f1dmvrnfibi 8808 pwfi 8819 cantnfcl 9130 en2eqpr 9433 fzfi 13341 hasheni 13709 fz1isolem 13820 isercolllem2 15022 isercoll 15024 summolem2 15073 zsum 15075 prodmolem2 15289 zprod 15291 bitsf1 15795 simpgnsgd 19222 ovoliunlem1 24103 wlksnfi 27686 eupthfi 27984 eulerpartlemgs2 31638 derangenlem 32418 erdsze2lem2 32451 heicant 34942 |
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