zapne - Intuitionistic Logic Explorer

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Theorem zapne 9698

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 9628	. . 3
2		zcn 9628	. . 3
3		apne 8941	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2421	. . 3
6		ztri3or 9666	. . . . . 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 9627	. . . . 5
13		zre 9627	. . . . 5
14		reaplt 8906	. . . . . 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

Syntax hints:

wn 3

wi 4 $/\$ wa 104

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

wceq 1402

wcel 2209

wne 2420 class class class wbr 4125

cc 8167

cr 8168

clt 8350 # cap 8899

cz 9623

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 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 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286

This theorem 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 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-br 4126 df-opab 4188 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-inn 9284 df-n0 9543 df-z 9624

This theorem is referenced by: zltlen 9703 msqznn 9725 qapne 10018 qreccl 10021 seqf1oglem1 10934 nn0opthd 11138 fihashneq0 11211 nnabscl 11844 eftcl 12399 dvdsval2 12535 dvdscmulr 12565 dvdsmulcr 12566 fsumdvds 12587 divconjdvds 12594 gcdn0gt0 12733 lcmcllem 12823 lcmid 12836 3lcm2e6woprm 12842 6lcm4e12 12843 mulgcddvds 12850 divgcdcoprmex 12858 cncongr1 12859 cncongr2 12860 isprm3 12874 pcpremul 13050 pceu 13052 pcmul 13058 pcdiv 13059 pcqmul 13060 dvdsprmpweqle 13094 qexpz 13109 4sqlem11 13158 relogbval 15976 relogbzcl 15977 nnlogbexp 15984 logbgcd1irraplemexp 15993 lgslem1 16033 lgsdilem2 16069 lgsdi 16070 lgsne0 16071 lgseisen 16107