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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 485 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ≈ 𝐵) | |
2 | enfii 8729 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) | |
3 | 2 | ancoms 461 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin) |
4 | hashen 13701 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵)) | |
5 | 3, 4 | sylancom 590 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵)) |
6 | 1, 5 | mpbird 259 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (♯‘𝐴) = (♯‘𝐵)) |
7 | relen 8508 | . . . . 5 ⊢ Rel ≈ | |
8 | 7 | brrelex1i 5603 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
9 | enfi 8728 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
10 | 9 | notbid 320 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ¬ 𝐵 ∈ Fin)) |
11 | 10 | biimpar 480 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → ¬ 𝐴 ∈ Fin) |
12 | hashinf 13689 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
13 | 8, 11, 12 | syl2an2r 683 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) = +∞) |
14 | 7 | brrelex2i 5604 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
15 | hashinf 13689 | . . . 4 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞) | |
16 | 14, 15 | sylan 582 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞) |
17 | 13, 16 | eqtr4d 2859 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) = (♯‘𝐵)) |
18 | 6, 17 | pm2.61dan 811 | 1 ⊢ (𝐴 ≈ 𝐵 → (♯‘𝐴) = (♯‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 class class class wbr 5059 ‘cfv 6350 ≈ cen 8500 Fincfn 8503 +∞cpnf 10666 ♯chash 13684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-hash 13685 |
This theorem is referenced by: hashen1 13725 hashfn 13730 hashfz 13782 hashf1lem2 13808 ishashinf 13815 hashgcdeq 16120 ramub2 16344 ram0 16352 odhash 18693 odhash2 18694 odngen 18696 lsmhash 18825 znhash 20699 znunithash 20705 cyggic 20713 birthdaylem2 25524 lgsquadlem1 25950 lgsquadlem2 25951 lgsquadlem3 25952 wlknwwlksneqs 27662 numclwwlk1 28134 dimval 30996 dimvalfi 30997 dimkerim 31018 fedgmul 31022 eulerpart 31635 ballotlemro 31775 ballotlemfrc 31779 ballotlem8 31789 rp-isfinite5 39876 |
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