zapne - Intuitionistic Logic Explorer

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Theorem zapne 9482

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 9412	. . 3
2		zcn 9412	. . 3
3		apne 8731	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2379	. . 3
6		ztri3or 9450	. . . . . 6 $/\$ $\/$ $\/$
7		3orrot 987	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		3orass 984	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$ $\/$ $\/$ $\/$
10	6, 9	sylib 122	. . . . 5 $/\$ $\/$ $\/$
11	10	ord 726	. . . 4 $/\$ $\/$
12		zre 9411	. . . . 5
13		zre 9411	. . . . 5
14		reaplt 8696	. . . . . 6 $/\$ # $\/$
15		orcom 730	. . . . . 6 $\/$ $\/$
16	14, 15	bitrdi 196	. . . . 5 $/\$ # $\/$
17	12, 13, 16	syl2an 289	. . . 4 $/\$ # $\/$
18	11, 17	sylibrd 169	. . 3 $/\$ #
19	5, 18	biimtrid 152	. 2 $/\$ #
20	4, 19	impbid 129	1 $/\$ #

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710 $\/$ w3o 980

wceq 1373

wcel 2178

wne 2378 class class class wbr 4059

cc 7958

cr 7959

clt 8142 # cap 8689

cz 9407

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-br 4060 df-opab 4122 df-id 4358 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-iota 5251 df-fun 5292 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-inn 9072 df-n0 9331 df-z 9408

This theorem is referenced by: zltlen 9486 msqznn 9508 qapne 9795 qreccl 9798 seqf1oglem1 10701 nn0opthd 10904 fihashneq0 10976 nnabscl 11526 eftcl 12080 dvdsval2 12216 dvdscmulr 12246 dvdsmulcr 12247 fsumdvds 12268 divconjdvds 12275 gcdn0gt0 12414 lcmcllem 12504 lcmid 12517 3lcm2e6woprm 12523 6lcm4e12 12524 mulgcddvds 12531 divgcdcoprmex 12539 cncongr1 12540 cncongr2 12541 isprm3 12555 pcpremul 12731 pceu 12733 pcmul 12739 pcdiv 12740 pcqmul 12741 dvdsprmpweqle 12775 qexpz 12790 4sqlem11 12839 relogbval 15538 relogbzcl 15539 nnlogbexp 15546 logbgcd1irraplemexp 15555 lgslem1 15592 lgsdilem2 15628 lgsdi 15629 lgsne0 15630 lgseisen 15666