Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > hashen | Unicode version |
Description: Two finite sets have the same number of elements iff they are equinumerous. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
hashen | ♯ ♯ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6735 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | adantr 274 | . 2 |
4 | isfi 6735 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad2antlr 486 | . . 3 |
7 | simplrl 530 | . . . . 5 | |
8 | simprl 526 | . . . . 5 | |
9 | nneneq 6831 | . . . . 5 | |
10 | 7, 8, 9 | syl2anc 409 | . . . 4 |
11 | simplrr 531 | . . . . . 6 | |
12 | enen1 6814 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | simprr 527 | . . . . . 6 | |
15 | enen2 6815 | . . . . . 6 | |
16 | 14, 15 | syl 14 | . . . . 5 |
17 | 13, 16 | bitrd 187 | . . . 4 |
18 | 11 | ensymd 6757 | . . . . . . 7 |
19 | hashennn 10701 | . . . . . . 7 ♯ frec | |
20 | 7, 18, 19 | syl2anc 409 | . . . . . 6 ♯ frec |
21 | 14 | ensymd 6757 | . . . . . . 7 |
22 | hashennn 10701 | . . . . . . 7 ♯ frec | |
23 | 8, 21, 22 | syl2anc 409 | . . . . . 6 ♯ frec |
24 | 20, 23 | eqeq12d 2185 | . . . . 5 ♯ ♯ frec frec |
25 | 0zd 9211 | . . . . . . . 8 | |
26 | eqid 2170 | . . . . . . . 8 frec frec | |
27 | 25, 26 | frec2uzf1od 10349 | . . . . . . 7 frec |
28 | f1of1 5439 | . . . . . . 7 frec frec | |
29 | 27, 28 | syl 14 | . . . . . 6 frec |
30 | f1fveq 5748 | . . . . . 6 frec frec frec | |
31 | 29, 7, 8, 30 | syl12anc 1231 | . . . . 5 frec frec |
32 | 24, 31 | bitrd 187 | . . . 4 ♯ ♯ |
33 | 10, 17, 32 | 3bitr4rd 220 | . . 3 ♯ ♯ |
34 | 6, 33 | rexlimddv 2592 | . 2 ♯ ♯ |
35 | 3, 34 | rexlimddv 2592 | 1 ♯ ♯ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wrex 2449 class class class wbr 3987 cmpt 4048 com 4572 wf1 5193 wf1o 5195 cfv 5196 (class class class)co 5850 freccfrec 6366 cen 6712 cfn 6714 cc0 7761 c1 7762 caddc 7764 cz 9199 cuz 9474 ♯chash 10696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-recs 6281 df-frec 6367 df-er 6509 df-en 6715 df-dom 6716 df-fin 6717 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-n0 9123 df-z 9200 df-uz 9475 df-ihash 10697 |
This theorem is referenced by: hasheqf1o 10706 isfinite4im 10714 fihasheq0 10715 hashsng 10720 fihashen1 10721 fihashfn 10722 hashun 10727 hashfz 10743 hashxp 10748 mertenslemi1 11485 hashdvds 12162 crth 12165 phimullem 12166 eulerth 12174 |
Copyright terms: Public domain | W3C validator |