zapne - Intuitionistic Logic Explorer

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Theorem zapne 9233

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 9167	. . 3
2		zcn 9167	. . 3
3		apne 8493	. . 3 $/\$ #
4	1, 2, 3	syl2an 287	. 2 $/\$ #
5		df-ne 2328	. . 3
6		ztri3or 9205	. . . . . 6 $/\$ $\/$ $\/$
7		3orrot 969	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		3orass 966	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
9	7, 8	bitri 183	. . . . . 6 $\/$ $\/$ $\/$ $\/$
10	6, 9	sylib 121	. . . . 5 $/\$ $\/$ $\/$
11	10	ord 714	. . . 4 $/\$ $\/$
12		zre 9166	. . . . 5
13		zre 9166	. . . . 5
14		reaplt 8458	. . . . . 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 962

wceq 1335

wcel 2128

wne 2327 class class class wbr 3965

cc 7725

cr 7726

clt 7907 # cap 8451

cz 9162

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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-mulrcl 7826 ax-addcom 7827 ax-mulcom 7828 ax-addass 7829 ax-mulass 7830 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-1rid 7834 ax-0id 7835 ax-rnegex 7836 ax-precex 7837 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-apti 7842 ax-pre-ltadd 7843 ax-pre-mulgt0 7844

This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-iota 5134 df-fun 5171 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-reap 8445 df-ap 8452 df-inn 8829 df-n0 9086 df-z 9163

This theorem is referenced by: zltlen 9237 msqznn 9259 qapne 9543 qreccl 9546 nn0opthd 10591 fihashneq0 10664 nnabscl 10995 eftcl 11546 dvdsval2 11681 dvdscmulr 11710 dvdsmulcr 11711 divconjdvds 11735 gcdn0gt0 11856 lcmcllem 11938 lcmid 11951 3lcm2e6woprm 11957 6lcm4e12 11958 mulgcddvds 11965 divgcdcoprmex 11973 cncongr1 11974 cncongr2 11975 isprm3 11989 relogbval 13255 relogbzcl 13256 nnlogbexp 13263 logbgcd1irraplemexp 13272