zapne - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > zapne		Unicode version

Theorem zapne 9327

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 9258	. . 3
2		zcn 9258	. . 3
3		apne 8580	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2348	. . 3
6		ztri3or 9296	. . . . . 6 $/\$ $\/$ $\/$
7		3orrot 984	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		3orass 981	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$ $\/$ $\/$ $\/$
10	6, 9	sylib 122	. . . . 5 $/\$ $\/$ $\/$
11	10	ord 724	. . . 4 $/\$ $\/$
12		zre 9257	. . . . 5
13		zre 9257	. . . . 5
14		reaplt 8545	. . . . . 6 $/\$ # $\/$
15		orcom 728	. . . . . 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 708 $\/$ w3o 977

wceq 1353

wcel 2148

wne 2347 class class class wbr 4004

cc 7809

cr 7810

clt 7992 # cap 8538

cz 9253

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928

This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-iota 5179 df-fun 5219 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-inn 8920 df-n0 9177 df-z 9254

This theorem is referenced by: zltlen 9331 msqznn 9353 qapne 9639 qreccl 9642 nn0opthd 10702 fihashneq0 10774 nnabscl 11109 eftcl 11662 dvdsval2 11797 dvdscmulr 11827 dvdsmulcr 11828 divconjdvds 11855 gcdn0gt0 11979 lcmcllem 12067 lcmid 12080 3lcm2e6woprm 12086 6lcm4e12 12087 mulgcddvds 12094 divgcdcoprmex 12102 cncongr1 12103 cncongr2 12104 isprm3 12118 pcpremul 12293 pceu 12295 pcmul 12301 pcdiv 12302 pcqmul 12303 dvdsprmpweqle 12336 qexpz 12350 relogbval 14372 relogbzcl 14373 nnlogbexp 14380 logbgcd1irraplemexp 14389 lgslem1 14404 lgsdilem2 14440 lgsdi 14441 lgsne0 14442