zapne - Intuitionistic Logic Explorer

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

Theorem zapne 9149

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 9083	. . 3
2		zcn 9083	. . 3
3		apne 8409	. . 3 $/\$ #
4	1, 2, 3	syl2an 287	. 2 $/\$ #
5		df-ne 2310	. . 3
6		ztri3or 9121	. . . . . 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 9082	. . . . 5
13		zre 9082	. . . . 5
14		reaplt 8374	. . . . . 6 $/\$ # $\/$
15		orcom 718	. . . . . 6 $\/$ $\/$
16	14, 15	syl6bb 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 1332

wcel 1481

wne 2309 class class class wbr 3937

cc 7642

cr 7643

clt 7824 # cap 8367

cz 9078

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761

This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-inn 8745 df-n0 9002 df-z 9079

This theorem is referenced by: zltlen 9153 msqznn 9175 qapne 9458 qreccl 9461 nn0opthd 10500 fihashneq0 10573 nnabscl 10904 eftcl 11397 dvdsval2 11532 dvdscmulr 11558 dvdsmulcr 11559 divconjdvds 11583 gcdn0gt0 11702 lcmcllem 11784 lcmid 11797 3lcm2e6woprm 11803 6lcm4e12 11804 mulgcddvds 11811 divgcdcoprmex 11819 cncongr1 11820 cncongr2 11821 isprm3 11835 relogbval 13076 relogbzcl 13077 nnlogbexp 13084 logbgcd1irraplemexp 13093