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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 8419 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
2 | 1 | biimparc 472 | 1 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 class class class wbr 4844 ≈ cen 8193 Fincfn 8196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-er 7983 df-en 8197 df-fin 8200 |
This theorem is referenced by: domfi 8424 en1eqsn 8433 isfinite2 8461 xpfi 8474 fofinf1o 8484 cnvfi 8491 f1dmvrnfibi 8493 pwfi 8504 cantnfcl 8815 en2eqpr 9117 fzfi 13025 hasheni 13387 fz1isolem 13493 isercolllem2 14736 isercoll 14738 summolem2a 14786 summolem2 14787 zsum 14789 prodmolem2a 15000 prodmolem2 15001 zprod 15003 bitsf1 15502 orbsta2 18058 ovoliunlem1 23609 wlksnfi 27186 eupthfi 27548 eulerpartlemgs2 30957 derangenlem 31669 erdsze2lem2 31702 heicant 33932 |
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