zapne - Intuitionistic Logic Explorer

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

Theorem zapne 9326

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 9257	. . 3
2		zcn 9257	. . 3
3		apne 8579	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2348	. . 3
6		ztri3or 9295	. . . . . 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 9256	. . . . 5
13		zre 9256	. . . . 5
14		reaplt 8544	. . . . . 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 4003

cc 7808

cr 7809

clt 7991 # cap 8537

cz 9252

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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927

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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-inn 8919 df-n0 9176 df-z 9253

This theorem is referenced by: zltlen 9330 msqznn 9352 qapne 9638 qreccl 9641 nn0opthd 10701 fihashneq0 10773 nnabscl 11108 eftcl 11661 dvdsval2 11796 dvdscmulr 11826 dvdsmulcr 11827 divconjdvds 11854 gcdn0gt0 11978 lcmcllem 12066 lcmid 12079 3lcm2e6woprm 12085 6lcm4e12 12086 mulgcddvds 12093 divgcdcoprmex 12101 cncongr1 12102 cncongr2 12103 isprm3 12117 pcpremul 12292 pceu 12294 pcmul 12300 pcdiv 12301 pcqmul 12302 dvdsprmpweqle 12335 qexpz 12349 relogbval 14305 relogbzcl 14306 nnlogbexp 14313 logbgcd1irraplemexp 14322 lgslem1 14337 lgsdilem2 14373 lgsdi 14374 lgsne0 14375