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Mirrors > Home > ILE Home > Th. List > hashun | Unicode version |
Description: The size of the union of disjoint finite sets is the sum of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
hashun | ♯ ♯ ♯ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6751 | . . . 4 | |
2 | 1 | biimpi 120 | . . 3 |
3 | 2 | 3ad2ant1 1018 | . 2 |
4 | isfi 6751 | . . . . . 6 | |
5 | 4 | biimpi 120 | . . . . 5 |
6 | 5 | 3ad2ant2 1019 | . . . 4 |
7 | 6 | adantr 276 | . . 3 |
8 | simplrl 535 | . . . . . 6 | |
9 | simprl 529 | . . . . . 6 | |
10 | eqid 2175 | . . . . . . 7 frec frec | |
11 | 10 | omgadd 10749 | . . . . . 6 frec frec frec |
12 | 8, 9, 11 | syl2anc 411 | . . . . 5 frec frec frec |
13 | nnacl 6471 | . . . . . . 7 | |
14 | 8, 9, 13 | syl2anc 411 | . . . . . 6 |
15 | enrefg 6754 | . . . . . . 7 | |
16 | 14, 15 | syl 14 | . . . . . 6 |
17 | hashennn 10726 | . . . . . 6 ♯ frec | |
18 | 14, 16, 17 | syl2anc 411 | . . . . 5 ♯ frec |
19 | vex 2738 | . . . . . . . 8 | |
20 | 19 | enref 6755 | . . . . . . 7 |
21 | hashennn 10726 | . . . . . . 7 ♯ frec | |
22 | 8, 20, 21 | sylancl 413 | . . . . . 6 ♯ frec |
23 | vex 2738 | . . . . . . . 8 | |
24 | 23 | enref 6755 | . . . . . . 7 |
25 | hashennn 10726 | . . . . . . 7 ♯ frec | |
26 | 9, 24, 25 | sylancl 413 | . . . . . 6 ♯ frec |
27 | 22, 26 | oveq12d 5883 | . . . . 5 ♯ ♯ frec frec |
28 | 12, 18, 27 | 3eqtr4d 2218 | . . . 4 ♯ ♯ ♯ |
29 | simpll1 1036 | . . . . . 6 | |
30 | simpll2 1037 | . . . . . 6 | |
31 | simpll3 1038 | . . . . . 6 | |
32 | simplrr 536 | . . . . . 6 | |
33 | simprr 531 | . . . . . 6 | |
34 | 29, 30, 31, 8, 9, 32, 33 | hashunlem 10751 | . . . . 5 |
35 | unfidisj 6911 | . . . . . . 7 | |
36 | 35 | ad2antrr 488 | . . . . . 6 |
37 | nnfi 6862 | . . . . . . . 8 | |
38 | 13, 37 | syl 14 | . . . . . . 7 |
39 | 8, 9, 38 | syl2anc 411 | . . . . . 6 |
40 | hashen 10730 | . . . . . 6 ♯ ♯ | |
41 | 36, 39, 40 | syl2anc 411 | . . . . 5 ♯ ♯ |
42 | 34, 41 | mpbird 167 | . . . 4 ♯ ♯ |
43 | nnfi 6862 | . . . . . . . 8 | |
44 | 8, 43 | syl 14 | . . . . . . 7 |
45 | hashen 10730 | . . . . . . 7 ♯ ♯ | |
46 | 29, 44, 45 | syl2anc 411 | . . . . . 6 ♯ ♯ |
47 | 32, 46 | mpbird 167 | . . . . 5 ♯ ♯ |
48 | nnfi 6862 | . . . . . . . 8 | |
49 | 9, 48 | syl 14 | . . . . . . 7 |
50 | hashen 10730 | . . . . . . 7 ♯ ♯ | |
51 | 30, 49, 50 | syl2anc 411 | . . . . . 6 ♯ ♯ |
52 | 33, 51 | mpbird 167 | . . . . 5 ♯ ♯ |
53 | 47, 52 | oveq12d 5883 | . . . 4 ♯ ♯ ♯ ♯ |
54 | 28, 42, 53 | 3eqtr4d 2218 | . . 3 ♯ ♯ ♯ |
55 | 7, 54 | rexlimddv 2597 | . 2 ♯ ♯ ♯ |
56 | 3, 55 | rexlimddv 2597 | 1 ♯ ♯ ♯ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 w3a 978 wceq 1353 wcel 2146 wrex 2454 cun 3125 cin 3126 c0 3420 class class class wbr 3998 cmpt 4059 com 4583 cfv 5208 (class class class)co 5865 freccfrec 6381 coa 6404 cen 6728 cfn 6730 cc0 7786 c1 7787 caddc 7789 cz 9224 ♯chash 10721 |
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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-frec 6382 df-1o 6407 df-oadd 6411 df-er 6525 df-en 6731 df-dom 6732 df-fin 6733 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-inn 8891 df-n0 9148 df-z 9225 df-uz 9500 df-ihash 10722 |
This theorem is referenced by: hashunsng 10754 fihashssdif 10765 hashxp 10773 fsumconst 11429 phiprmpw 12188 |
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