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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 484 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ≈ 𝐵) | |
2 | enfii 9184 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) | |
3 | 2 | ancoms 460 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin) |
4 | hashen 14302 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵)) | |
5 | 3, 4 | sylancom 589 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵)) |
6 | 1, 5 | mpbird 257 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (♯‘𝐴) = (♯‘𝐵)) |
7 | relen 8939 | . . . . 5 ⊢ Rel ≈ | |
8 | 7 | brrelex1i 5729 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
9 | enfi 9185 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
10 | 9 | notbid 318 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ¬ 𝐵 ∈ Fin)) |
11 | 10 | biimpar 479 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → ¬ 𝐴 ∈ Fin) |
12 | hashinf 14290 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
13 | 8, 11, 12 | syl2an2r 684 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) = +∞) |
14 | 7 | brrelex2i 5730 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
15 | hashinf 14290 | . . . 4 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞) | |
16 | 14, 15 | sylan 581 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞) |
17 | 13, 16 | eqtr4d 2776 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) = (♯‘𝐵)) |
18 | 6, 17 | pm2.61dan 812 | 1 ⊢ (𝐴 ≈ 𝐵 → (♯‘𝐴) = (♯‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 class class class wbr 5146 ‘cfv 6539 ≈ cen 8931 Fincfn 8934 +∞cpnf 11240 ♯chash 14285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-n0 12468 df-z 12554 df-uz 12818 df-hash 14286 |
This theorem is referenced by: hashen1 14325 hashfn 14330 hashfz 14382 hashf1lem2 14412 ishashinf 14419 hashgcdeq 16717 ramub2 16942 ram0 16950 odhash 19434 odhash2 19435 odngen 19437 znhash 21097 znunithash 21103 cyggic 21111 birthdaylem2 26436 lgsquadlem1 26862 lgsquadlem2 26863 lgsquadlem3 26864 wlknwwlksneqs 29123 numclwwlk1 29593 dimval 32631 dimvalfi 32632 dimkerim 32656 fedgmul 32660 eulerpart 33318 ballotlemro 33458 ballotlemfrc 33462 ballotlem8 33472 sticksstones5 40903 sticksstones20 40919 rp-isfinite5 42200 |
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