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Mirrors > Home > ILE Home > Th. List > hashcl | Unicode version |
Description: Closure of the ♯ function. (Contributed by Paul Chapman, 26-Oct-2012.) (Revised by Mario Carneiro, 13-Jul-2014.) |
Ref | Expression |
---|---|
hashcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6663 |
. . 3
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2 | 1 | biimpi 119 |
. 2
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3 | simprl 521 |
. . . 4
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4 | simprr 522 |
. . . . 5
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5 | 4 | ensymd 6685 |
. . . 4
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6 | hashennn 10558 |
. . . 4
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7 | 3, 5, 6 | syl2anc 409 |
. . 3
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8 | 0zd 9090 |
. . . . . 6
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9 | eqid 2140 |
. . . . . 6
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10 | id 19 |
. . . . . 6
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11 | 8, 9, 10 | frec2uzuzd 10206 |
. . . . 5
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12 | nn0uz 9384 |
. . . . 5
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13 | 11, 12 | eleqtrrdi 2234 |
. . . 4
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14 | 3, 13 | syl 14 |
. . 3
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15 | 7, 14 | eqeltrd 2217 |
. 2
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16 | 2, 15 | rexlimddv 2557 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-recs 6210 df-frec 6296 df-er 6437 df-en 6643 df-dom 6644 df-fin 6645 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-ihash 10554 |
This theorem is referenced by: hashfiv01gt1 10560 filtinf 10570 isfinite4im 10571 fihashneq0 10573 hashnncl 10574 fihashssdif 10596 hashdifpr 10598 hashxp 10604 zfz1isolemsplit 10613 zfz1isolemiso 10614 zfz1isolem1 10615 fz1f1o 11176 fsumconst 11255 hashiun 11279 hash2iun1dif1 11281 phival 11925 phicl2 11926 phiprmpw 11934 |
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