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Mirrors > Home > MPE Home > Th. List > enfi | Structured version Visualization version GIF version |
Description: Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.) |
Ref | Expression |
---|---|
enfi | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enen1 8100 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≈ 𝑥 ↔ 𝐵 ≈ 𝑥)) | |
2 | 1 | rexbidv 3052 | . 2 ⊢ (𝐴 ≈ 𝐵 → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)) |
3 | isfi 7979 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
4 | isfi 7979 | . 2 ⊢ (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) | |
5 | 2, 3, 4 | 3bitr4g 303 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 ∃wrex 2913 class class class wbr 4653 ωcom 7065 ≈ cen 7952 Fincfn 7955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-er 7742 df-en 7956 df-fin 7959 |
This theorem is referenced by: enfii 8177 wofib 8450 en2eleq 8831 sdom2en01 9124 fin23lem21 9161 enfin1ai 9206 fin17 9216 isfin7-2 9218 engch 9450 uzinf 12764 hasheni 13136 isfinite4 13153 symggen 17890 psgnunilem1 17913 dfod2 17981 odhash 17989 gsumval3lem1 18306 gsumval3lem2 18307 gsumval3 18308 cyggic 19921 cusgrfilem3 26353 derangen 31154 erdsze2lem1 31185 phpreu 33393 lindsdom 33403 poimirlem30 33439 diophin 37336 diophren 37377 fiphp3d 37383 fiuneneq 37775 |
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