zapne - Intuitionistic Logic Explorer

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Theorem zapne 9728

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 9658	. . 3
2		zcn 9658	. . 3
3		apne 8959	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2421	. . 3
6		ztri3or 9696	. . . . . 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 9657	. . . . 5
13		zre 9657	. . . . 5
14		reaplt 8924	. . . . . 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 4131

cc 8183

cr 8184

clt 8366 # cap 8917

cz 9653

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 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302

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 4132 df-opab 4194 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-iota 5340 df-fun 5382 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-inn 9314 df-n0 9575 df-z 9654

This theorem is used by: zltlen 9733 msqznn 9756 qapne 10054 qreccl 10057 seqf1oglem1 10976 nn0opthd 11181 fihashneq0 11254 nnabscl 11888 eftcl 12445 dvdsval2 12581 dvdscmulr 12611 dvdsmulcr 12612 fsumdvds 12633 divconjdvds 12640 gcdn0gt0 12779 lcmcllem 12869 lcmid 12882 3lcm2e6woprm 12888 6lcm4e12 12889 mulgcddvds 12896 divgcdcoprmex 12904 cncongr1 12905 cncongr2 12906 isprm3 12920 pcpremul 13100 pceu 13102 pcmul 13108 pcdiv 13109 pcqmul 13110 dvdsprmpweqle 13144 qexpz 13159 4sqlem11 13208 relogbval 16204 relogbzcl 16205 nnlogbexp 16212 logbgcd1irraplemexp 16221 lgslem1 16341 lgsdilem2 16377 lgsdi 16378 lgsne0 16379 lgseisen 16415