Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7985 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≈ 𝑥 ↔ 𝐵 ≈ 𝑥)) | |
2 | 1 | rexbidv 3034 | . 2 ⊢ (𝐴 ≈ 𝐵 → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)) |
3 | isfi 7865 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
4 | isfi 7865 | . 2 ⊢ (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) | |
5 | 2, 3, 4 | 3bitr4g 302 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ωcom 6957 ≈ cen 7838 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-er 7629 df-en 7842 df-fin 7845 |
This theorem is referenced by: enfii 8062 wofib 8333 en2eleq 8714 sdom2en01 9007 fin23lem21 9044 enfin1ai 9089 fin17 9099 isfin7-2 9101 engch 9329 uzinf 12626 hasheni 12998 isfinite4 13014 symggen 17713 psgnunilem1 17736 dfod2 17804 odhash 17812 gsumval3lem1 18129 gsumval3lem2 18130 gsumval3 18131 cyggic 19740 nbusgrafi 25977 cusgrafilem3 26009 eupai 26494 derangen 30408 erdsze2lem1 30439 phpreu 32563 lindsdom 32573 poimirlem30 32609 diophin 36354 diophren 36395 fiphp3d 36401 fiuneneq 36794 cusgrfilem3 40673 |
Copyright terms: Public domain | W3C validator |