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Mirrors > Home > MPE Home > Th. List > hashprg | Structured version Visualization version Unicode version |
Description: The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) |
Ref | Expression |
---|---|
hashprg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 463 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | elsni 3995 |
. . . . . . 7
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3 | 2 | eqcomd 2459 |
. . . . . 6
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4 | 3 | necon3ai 2651 |
. . . . 5
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5 | snfi 7655 |
. . . . . 6
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6 | hashunsng 12578 |
. . . . . . 7
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7 | 6 | imp 431 |
. . . . . 6
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8 | 5, 7 | mpanr1 690 |
. . . . 5
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9 | 1, 4, 8 | syl2an 480 |
. . . 4
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10 | hashsng 12556 |
. . . . . . 7
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11 | 10 | adantr 467 |
. . . . . 6
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12 | 11 | adantr 467 |
. . . . 5
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13 | 12 | oveq1d 6310 |
. . . 4
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14 | 9, 13 | eqtrd 2487 |
. . 3
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15 | df-pr 3973 |
. . . 4
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16 | 15 | fveq2i 5873 |
. . 3
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17 | df-2 10675 |
. . 3
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18 | 14, 16, 17 | 3eqtr4g 2512 |
. 2
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19 | 1ne2 10829 |
. . . . . . 7
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20 | 19 | a1i 11 |
. . . . . 6
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21 | 11, 20 | eqnetrd 2693 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | dfsn2 3983 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | preq2 4055 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 22, 23 | syl5req 2500 |
. . . . . . 7
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25 | 24 | fveq2d 5874 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 25 | neeq1d 2685 |
. . . . 5
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27 | 21, 26 | syl5ibrcom 226 |
. . . 4
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28 | 27 | necon2d 2649 |
. . 3
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29 | 28 | imp 431 |
. 2
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30 | 18, 29 | impbida 844 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-int 4238 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-1st 6798 df-2nd 6799 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-1o 7187 df-oadd 7191 df-er 7368 df-en 7575 df-dom 7576 df-sdom 7577 df-fin 7578 df-card 8378 df-cda 8603 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 df-nn 10617 df-2 10675 df-n0 10877 df-z 10945 df-uz 11167 df-fz 11792 df-hash 12523 |
This theorem is referenced by: hashprb 12581 prhash2ex 12583 hashfun 12616 hash2exprb 12639 hashtpg 12648 elss2prb 12650 wrdlen2i 13033 prmreclem2 14873 isnzr2hash 18500 dchrisum0re 24363 nehash2 24564 umgraex 25062 usgra1 25112 usgranloopv 25117 usgraexmplef 25140 cusgraexi 25208 cusgrafilem1 25219 2trllemA 25292 2pthon 25344 2pthon3v 25346 nbhashuvtx1 25655 eupath 25721 konigsberg 25727 coinflipprob 29324 subfacp1lem1 29914 poimirlem9 31961 fourierdlem54 38034 fourierdlem102 38082 fourierdlem103 38083 fourierdlem104 38084 fourierdlem114 38094 upgrex 39194 usgr1e 39330 cusgrexi 39517 cusgrfilem1 39526 umgr2v2e 39572 usgpredgdv 39825 |
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