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Mirrors > Home > ILE Home > Th. List > zapne | Unicode version |
Description: Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
zapne | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9052 | . . 3 | |
2 | zcn 9052 | . . 3 | |
3 | apne 8378 | . . 3 # | |
4 | 1, 2, 3 | syl2an 287 | . 2 # |
5 | df-ne 2307 | . . 3 | |
6 | ztri3or 9090 | . . . . . 6 | |
7 | 3orrot 968 | . . . . . . 7 | |
8 | 3orass 965 | . . . . . . 7 | |
9 | 7, 8 | bitri 183 | . . . . . 6 |
10 | 6, 9 | sylib 121 | . . . . 5 |
11 | 10 | ord 713 | . . . 4 |
12 | zre 9051 | . . . . 5 | |
13 | zre 9051 | . . . . 5 | |
14 | reaplt 8343 | . . . . . 6 # | |
15 | orcom 717 | . . . . . 6 | |
16 | 14, 15 | syl6bb 195 | . . . . 5 # |
17 | 12, 13, 16 | syl2an 287 | . . . 4 # |
18 | 11, 17 | sylibrd 168 | . . 3 # |
19 | 5, 18 | syl5bi 151 | . 2 # |
20 | 4, 19 | impbid 128 | 1 # |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3o 961 wceq 1331 wcel 1480 wne 2306 class class class wbr 3924 cc 7611 cr 7612 clt 7793 # cap 8336 cz 9047 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: zltlen 9122 msqznn 9144 qapne 9424 qreccl 9427 nn0opthd 10461 fihashneq0 10534 nnabscl 10865 eftcl 11349 dvdsval2 11485 dvdscmulr 11511 dvdsmulcr 11512 divconjdvds 11536 gcdn0gt0 11655 lcmcllem 11737 lcmid 11750 3lcm2e6woprm 11756 6lcm4e12 11757 mulgcddvds 11764 divgcdcoprmex 11772 cncongr1 11773 cncongr2 11774 isprm3 11788 |
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