zapne - Intuitionistic Logic Explorer

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Theorem zapne 9521

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 9451	. . 3
2		zcn 9451	. . 3
3		apne 8770	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2401	. . 3
6		ztri3or 9489	. . . . . 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 9450	. . . . 5
13		zre 9450	. . . . 5
14		reaplt 8735	. . . . . 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 7997

cr 7998

clt 8181 # cap 8728

cz 9446

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 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116

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 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-inn 9111 df-n0 9370 df-z 9447

This theorem is referenced by: zltlen 9525 msqznn 9547 qapne 9834 qreccl 9837 seqf1oglem1 10741 nn0opthd 10944 fihashneq0 11016 nnabscl 11611 eftcl 12165 dvdsval2 12301 dvdscmulr 12331 dvdsmulcr 12332 fsumdvds 12353 divconjdvds 12360 gcdn0gt0 12499 lcmcllem 12589 lcmid 12602 3lcm2e6woprm 12608 6lcm4e12 12609 mulgcddvds 12616 divgcdcoprmex 12624 cncongr1 12625 cncongr2 12626 isprm3 12640 pcpremul 12816 pceu 12818 pcmul 12824 pcdiv 12825 pcqmul 12826 dvdsprmpweqle 12860 qexpz 12875 4sqlem11 12924 relogbval 15625 relogbzcl 15626 nnlogbexp 15633 logbgcd1irraplemexp 15642 lgslem1 15679 lgsdilem2 15715 lgsdi 15716 lgsne0 15717 lgseisen 15753