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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6787 |
. . . 4
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2 | 1 | biimpi 120 |
. . 3
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3 | 2 | adantr 276 |
. 2
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4 | isfi 6787 |
. . . . 5
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5 | 4 | biimpi 120 |
. . . 4
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6 | 5 | ad2antlr 489 |
. . 3
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7 | simplrl 535 |
. . . . 5
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8 | simprl 529 |
. . . . 5
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9 | nneneq 6885 |
. . . . 5
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10 | 7, 8, 9 | syl2anc 411 |
. . . 4
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11 | simplrr 536 |
. . . . . 6
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12 | enen1 6868 |
. . . . . 6
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13 | 11, 12 | syl 14 |
. . . . 5
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14 | simprr 531 |
. . . . . 6
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15 | enen2 6869 |
. . . . . 6
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16 | 14, 15 | syl 14 |
. . . . 5
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17 | 13, 16 | bitrd 188 |
. . . 4
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18 | 11 | ensymd 6809 |
. . . . . . 7
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19 | hashennn 10792 |
. . . . . . 7
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20 | 7, 18, 19 | syl2anc 411 |
. . . . . 6
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21 | 14 | ensymd 6809 |
. . . . . . 7
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22 | hashennn 10792 |
. . . . . . 7
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23 | 8, 21, 22 | syl2anc 411 |
. . . . . 6
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24 | 20, 23 | eqeq12d 2204 |
. . . . 5
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25 | 0zd 9295 |
. . . . . . . 8
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26 | eqid 2189 |
. . . . . . . 8
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27 | 25, 26 | frec2uzf1od 10437 |
. . . . . . 7
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28 | f1of1 5479 |
. . . . . . 7
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29 | 27, 28 | syl 14 |
. . . . . 6
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30 | f1fveq 5794 |
. . . . . 6
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31 | 29, 7, 8, 30 | syl12anc 1247 |
. . . . 5
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32 | 24, 31 | bitrd 188 |
. . . 4
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33 | 10, 17, 32 | 3bitr4rd 221 |
. . 3
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34 | 6, 33 | rexlimddv 2612 |
. 2
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35 | 3, 34 | rexlimddv 2612 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-recs 6330 df-frec 6416 df-er 6559 df-en 6767 df-dom 6768 df-fin 6769 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-inn 8950 df-n0 9207 df-z 9284 df-uz 9559 df-ihash 10788 |
This theorem is referenced by: hasheqf1o 10797 isfinite4im 10804 fihasheq0 10805 hashsng 10810 fihashen1 10811 fihashfn 10812 hashun 10817 hashfz 10833 hashxp 10838 mertenslemi1 11575 hashdvds 12253 crth 12256 phimullem 12257 eulerth 12265 4sqlem11 12433 |
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