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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 486 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ≈ 𝐵) | |
2 | enfii 8864 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin) | |
3 | 2 | ancoms 462 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin) |
4 | hashen 13913 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵)) | |
5 | 3, 4 | sylancom 591 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) = (♯‘𝐵) ↔ 𝐴 ≈ 𝐵)) |
6 | 1, 5 | mpbird 260 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (♯‘𝐴) = (♯‘𝐵)) |
7 | relen 8631 | . . . . 5 ⊢ Rel ≈ | |
8 | 7 | brrelex1i 5605 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V) |
9 | enfi 8865 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
10 | 9 | notbid 321 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ¬ 𝐵 ∈ Fin)) |
11 | 10 | biimpar 481 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → ¬ 𝐴 ∈ Fin) |
12 | hashinf 13901 | . . . 4 ⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
13 | 8, 11, 12 | syl2an2r 685 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) = +∞) |
14 | 7 | brrelex2i 5606 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V) |
15 | hashinf 13901 | . . . 4 ⊢ ((𝐵 ∈ V ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞) | |
16 | 14, 15 | sylan 583 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞) |
17 | 13, 16 | eqtr4d 2780 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) = (♯‘𝐵)) |
18 | 6, 17 | pm2.61dan 813 | 1 ⊢ (𝐴 ≈ 𝐵 → (♯‘𝐴) = (♯‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 class class class wbr 5053 ‘cfv 6380 ≈ cen 8623 Fincfn 8626 +∞cpnf 10864 ♯chash 13896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-hash 13897 |
This theorem is referenced by: hashen1 13937 hashfn 13942 hashfz 13994 hashf1lem2 14022 ishashinf 14029 hashgcdeq 16342 ramub2 16567 ram0 16575 odhash 18963 odhash2 18964 odngen 18966 znhash 20523 znunithash 20529 cyggic 20537 birthdaylem2 25835 lgsquadlem1 26261 lgsquadlem2 26262 lgsquadlem3 26263 wlknwwlksneqs 27974 numclwwlk1 28444 dimval 31400 dimvalfi 31401 dimkerim 31422 fedgmul 31426 eulerpart 32061 ballotlemro 32201 ballotlemfrc 32205 ballotlem8 32215 sticksstones5 39828 sticksstones20 39844 rp-isfinite5 40809 |
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