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Mirrors > Home > MPE Home > Th. List > enfi | Structured version Visualization version GIF version |
Description: Equinumerous sets have the same finiteness. For a shorter proof using ax-pow 5363, see enfiALT 9197. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5363. (Revised by BTernaryTau, 23-Sep-2024.) |
Ref | Expression |
---|---|
enfi | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensymfib 9193 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴)) | |
2 | 1 | pm5.32i 574 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵) ↔ (𝐴 ∈ Fin ∧ 𝐵 ≈ 𝐴)) |
3 | enfii 9195 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≈ 𝐴) → 𝐵 ∈ Fin) | |
4 | 2, 3 | sylbi 216 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐵 ∈ Fin) |
5 | 4 | expcom 413 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin → 𝐵 ∈ Fin)) |
6 | enfii 9195 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) | |
7 | 6 | expcom 413 | . 2 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ Fin → 𝐴 ∈ Fin)) |
8 | 5, 7 | impbid 211 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2105 class class class wbr 5148 ≈ cen 8942 Fincfn 8945 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7860 df-1o 8472 df-en 8946 df-fin 8949 |
This theorem is referenced by: enfiiOLD 9270 wofib 9546 en2eleq 10009 sdom2en01 10303 fin23lem21 10340 enfin1ai 10385 fin17 10395 isfin7-2 10397 engch 10629 uzinf 13937 hasheni 14315 isfinite4 14329 symggen 19386 psgnunilem1 19409 dfod2 19480 odhash 19490 gsumval3lem2 19822 gsumval3 19823 cyggic 21438 cusgrfilem3 29148 unidifsnel 32206 unidifsnne 32207 derangen 34628 erdsze2lem1 34659 phpreu 36938 lindsdom 36948 poimirlem30 36984 diophin 41975 diophren 42016 fiphp3d 42022 fiuneneq 42404 |
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