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 8657 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ≈ 𝑥 ↔ 𝐵 ≈ 𝑥)) | |
2 | 1 | rexbidv 3297 | . 2 ⊢ (𝐴 ≈ 𝐵 → (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥)) |
3 | isfi 8533 | . 2 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥) | |
4 | isfi 8533 | . 2 ⊢ (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵 ≈ 𝑥) | |
5 | 2, 3, 4 | 3bitr4g 316 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 ∃wrex 3139 class class class wbr 5066 ωcom 7580 ≈ 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: enfii 8735 wofib 9009 en2eleq 9434 sdom2en01 9724 fin23lem21 9761 enfin1ai 9806 fin17 9816 isfin7-2 9818 engch 10050 uzinf 13334 hasheni 13709 isfinite4 13724 symggen 18598 psgnunilem1 18621 dfod2 18691 odhash 18699 gsumval3lem2 19026 gsumval3 19027 cyggic 20719 cusgrfilem3 27239 unidifsnel 30295 unidifsnne 30296 derangen 32419 erdsze2lem1 32450 phpreu 34891 lindsdom 34901 poimirlem30 34937 diophin 39389 diophren 39430 fiphp3d 39436 fiuneneq 39817 |
Copyright terms: Public domain | W3C validator |