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Mirrors > Home > MPE Home > Th. List > hasheni | Structured version Visualization version GIF version |
Description: Equinumerous sets have the same number of elements (even if they are not finite). (Contributed by Mario Carneiro, 15-Apr-2015.) |
Ref | Expression |
---|---|
hasheni | ⊢ (𝐴 ≈ 𝐵 → (♯‘𝐴) = (♯‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ≈ 𝐵) | |
2 | enfii 9191 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) | |
3 | 2 | ancoms 457 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin) |
4 | hashen 14311 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵)) | |
5 | 3, 4 | sylancom 586 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵)) |
6 | 1, 5 | mpbird 256 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (♯‘𝐴) = (♯‘𝐵)) |
7 | relen 8946 | . . . . 5 ⊢ Rel ≈ | |
8 | 7 | brrelex1i 5731 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
9 | enfi 9192 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
10 | 9 | notbid 317 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ¬ 𝐵 ∈ Fin)) |
11 | 10 | biimpar 476 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → ¬ 𝐴 ∈ Fin) |
12 | hashinf 14299 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
13 | 8, 11, 12 | syl2an2r 681 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) = +∞) |
14 | 7 | brrelex2i 5732 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
15 | hashinf 14299 | . . . 4 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞) | |
16 | 14, 15 | sylan 578 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞) |
17 | 13, 16 | eqtr4d 2773 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) = (♯‘𝐵)) |
18 | 6, 17 | pm2.61dan 809 | 1 ⊢ (𝐴 ≈ 𝐵 → (♯‘𝐴) = (♯‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 class class class wbr 5147 ‘cfv 6542 ≈ cen 8938 Fincfn 8941 +∞cpnf 11249 ♯chash 14294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-hash 14295 |
This theorem is referenced by: hashen1 14334 hashfn 14339 hashfz 14391 hashf1lem2 14421 ishashinf 14428 hashgcdeq 16726 ramub2 16951 ram0 16959 odhash 19483 odhash2 19484 odngen 19486 znhash 21333 znunithash 21339 cyggic 21347 birthdaylem2 26693 lgsquadlem1 27119 lgsquadlem2 27120 lgsquadlem3 27121 wlknwwlksneqs 29411 numclwwlk1 29881 dimval 32973 dimvalfi 32974 dimkerim 33000 fedgmul 33004 eulerpart 33679 ballotlemro 33819 ballotlemfrc 33823 ballotlem8 33833 sticksstones5 41272 sticksstones20 41288 rp-isfinite5 42570 |
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