zapne - Intuitionistic Logic Explorer

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Theorem zapne 9723

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 9653	. . 3
2		zcn 9653	. . 3
3		apne 8953	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2421	. . 3
6		ztri3or 9691	. . . . . 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 9652	. . . . 5
13		zre 9652	. . . . 5
14		reaplt 8918	. . . . . 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 8911

cz 9648

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 8500 df-neg 8501 df-reap 8905 df-ap 8912 df-inn 9307 df-n0 9568 df-z 9649

This theorem is used by: zltlen 9728 msqznn 9750 qapne 10048 qreccl 10051 seqf1oglem1 10969 nn0opthd 11174 fihashneq0 11247 nnabscl 11881 eftcl 12437 dvdsval2 12573 dvdscmulr 12603 dvdsmulcr 12604 fsumdvds 12625 divconjdvds 12632 gcdn0gt0 12771 lcmcllem 12861 lcmid 12874 3lcm2e6woprm 12880 6lcm4e12 12881 mulgcddvds 12888 divgcdcoprmex 12896 cncongr1 12897 cncongr2 12898 isprm3 12912 pcpremul 13092 pceu 13094 pcmul 13100 pcdiv 13101 pcqmul 13102 dvdsprmpweqle 13136 qexpz 13151 4sqlem11 13200 relogbval 16106 relogbzcl 16107 nnlogbexp 16114 logbgcd1irraplemexp 16123 lgslem1 16217 lgsdilem2 16253 lgsdi 16254 lgsne0 16255 lgseisen 16291