zapne - Intuitionistic Logic Explorer

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

Theorem zapne 9532

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 9462	. . 3
2		zcn 9462	. . 3
3		apne 8781	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2401	. . 3
6		ztri3or 9500	. . . . . 6 $/\$ $\/$ $\/$
7		3orrot 1008	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		3orass 1005	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$ $\/$ $\/$ $\/$
10	6, 9	sylib 122	. . . . 5 $/\$ $\/$ $\/$
11	10	ord 729	. . . 4 $/\$ $\/$
12		zre 9461	. . . . 5
13		zre 9461	. . . . 5
14		reaplt 8746	. . . . . 6 $/\$ # $\/$
15		orcom 733	. . . . . 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 713 $\/$ w3o 1001

wceq 1395

wcel 2200

wne 2400 class class class wbr 4083

cc 8008

cr 8009

clt 8192 # cap 8739

cz 9457

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-inn 9122 df-n0 9381 df-z 9458

This theorem is referenced by: zltlen 9536 msqznn 9558 qapne 9846 qreccl 9849 seqf1oglem1 10753 nn0opthd 10956 fihashneq0 11028 nnabscl 11627 eftcl 12181 dvdsval2 12317 dvdscmulr 12347 dvdsmulcr 12348 fsumdvds 12369 divconjdvds 12376 gcdn0gt0 12515 lcmcllem 12605 lcmid 12618 3lcm2e6woprm 12624 6lcm4e12 12625 mulgcddvds 12632 divgcdcoprmex 12640 cncongr1 12641 cncongr2 12642 isprm3 12656 pcpremul 12832 pceu 12834 pcmul 12840 pcdiv 12841 pcqmul 12842 dvdsprmpweqle 12876 qexpz 12891 4sqlem11 12940 relogbval 15641 relogbzcl 15642 nnlogbexp 15649 logbgcd1irraplemexp 15658 lgslem1 15695 lgsdilem2 15731 lgsdi 15732 lgsne0 15733 lgseisen 15769