zapne - Intuitionistic Logic Explorer

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Theorem zapne 9719

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 9649	. . 3
2		zcn 9649	. . 3
3		apne 8951	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2421	. . 3
6		ztri3or 9687	. . . . . 6 $/\$ $\/$ $\/$
7		3orrot 1015	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		3orass 1012	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$ $\/$ $\/$ $\/$
10	6, 9	sylib 122	. . . . 5 $/\$ $\/$ $\/$
11	10	ord 736	. . . 4 $/\$ $\/$
12		zre 9648	. . . . 5
13		zre 9648	. . . . 5
14		reaplt 8916	. . . . . 6 $/\$ # $\/$
15		orcom 740	. . . . . 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

This proof depends on syntax axioms:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 $\/$ w3o 1008

wceq 1402

wcel 2209

wne 2420 class class class wbr 4130

cc 8177

cr 8178

clt 8360 # cap 8909

cz 9644

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296

This proof depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-inn 9305 df-n0 9564 df-z 9645

This theorem is used by: zltlen 9724 msqznn 9746 qapne 10039 qreccl 10042 seqf1oglem1 10956 nn0opthd 11160 fihashneq0 11233 nnabscl 11866 eftcl 12421 dvdsval2 12557 dvdscmulr 12587 dvdsmulcr 12588 fsumdvds 12609 divconjdvds 12616 gcdn0gt0 12755 lcmcllem 12845 lcmid 12858 3lcm2e6woprm 12864 6lcm4e12 12865 mulgcddvds 12872 divgcdcoprmex 12880 cncongr1 12881 cncongr2 12882 isprm3 12896 pcpremul 13072 pceu 13074 pcmul 13080 pcdiv 13081 pcqmul 13082 dvdsprmpweqle 13116 qexpz 13131 4sqlem11 13180 relogbval 16053 relogbzcl 16054 nnlogbexp 16061 logbgcd1irraplemexp 16070 lgslem1 16119 lgsdilem2 16155 lgsdi 16156 lgsne0 16157 lgseisen 16193