zapne - Intuitionistic Logic Explorer

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Theorem zapne 9261

Description: Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)

**Assertion**
Ref	Expression
zapne	$/\$ #

**Proof of Theorem zapne**
Step	Hyp	Ref	Expression
1		zcn 9192	. . 3
2		zcn 9192	. . 3
3		apne 8517	. . 3 $/\$ #
4	1, 2, 3	syl2an 287	. 2 $/\$ #
5		df-ne 2336	. . 3
6		ztri3or 9230	. . . . . 6 $/\$ $\/$ $\/$
7		3orrot 974	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		3orass 971	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
9	7, 8	bitri 183	. . . . . 6 $\/$ $\/$ $\/$ $\/$
10	6, 9	sylib 121	. . . . 5 $/\$ $\/$ $\/$
11	10	ord 714	. . . 4 $/\$ $\/$
12		zre 9191	. . . . 5
13		zre 9191	. . . . 5
14		reaplt 8482	. . . . . 6 $/\$ # $\/$
15		orcom 718	. . . . . 6 $\/$ $\/$
16	14, 15	bitrdi 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 698 $\/$ w3o 967

wceq 1343

wcel 2136

wne 2335 class class class wbr 3981

cc 7747

cr 7748

clt 7929 # cap 8475

cz 9187

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4099 ax-pow 4152 ax-pr 4186 ax-un 4410 ax-setind 4513 ax-cnex 7840 ax-resscn 7841 ax-1cn 7842 ax-1re 7843 ax-icn 7844 ax-addcl 7845 ax-addrcl 7846 ax-mulcl 7847 ax-mulrcl 7848 ax-addcom 7849 ax-mulcom 7850 ax-addass 7851 ax-mulass 7852 ax-distr 7853 ax-i2m1 7854 ax-0lt1 7855 ax-1rid 7856 ax-0id 7857 ax-rnegex 7858 ax-precex 7859 ax-cnre 7860 ax-pre-ltirr 7861 ax-pre-ltwlin 7862 ax-pre-lttrn 7863 ax-pre-apti 7864 ax-pre-ltadd 7865 ax-pre-mulgt0 7866

This theorem depends on definitions: df-bi 116 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-ne 2336 df-nel 2431 df-ral 2448 df-rex 2449 df-reu 2450 df-rab 2452 df-v 2727 df-sbc 2951 df-dif 3117 df-un 3119 df-in 3121 df-ss 3128 df-pw 3560 df-sn 3581 df-pr 3582 df-op 3584 df-uni 3789 df-int 3824 df-br 3982 df-opab 4043 df-id 4270 df-xp 4609 df-rel 4610 df-cnv 4611 df-co 4612 df-dm 4613 df-iota 5152 df-fun 5189 df-fv 5195 df-riota 5797 df-ov 5844 df-oprab 5845 df-mpo 5846 df-pnf 7931 df-mnf 7932 df-xr 7933 df-ltxr 7934 df-le 7935 df-sub 8067 df-neg 8068 df-reap 8469 df-ap 8476 df-inn 8854 df-n0 9111 df-z 9188

This theorem is referenced by: zltlen 9265 msqznn 9287 qapne 9573 qreccl 9576 nn0opthd 10631 fihashneq0 10704 nnabscl 11038 eftcl 11591 dvdsval2 11726 dvdscmulr 11756 dvdsmulcr 11757 divconjdvds 11783 gcdn0gt0 11907 lcmcllem 11995 lcmid 12008 3lcm2e6woprm 12014 6lcm4e12 12015 mulgcddvds 12022 divgcdcoprmex 12030 cncongr1 12031 cncongr2 12032 isprm3 12046 pcpremul 12221 pceu 12223 pcmul 12229 pcdiv 12230 pcqmul 12231 dvdsprmpweqle 12264 qexpz 12278 relogbval 13469 relogbzcl 13470 nnlogbexp 13477 logbgcd1irraplemexp 13486 lgslem1 13501 lgsdilem2 13537 lgsdi 13538 lgsne0 13539