zapne - Intuitionistic Logic Explorer

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Theorem zapne 9724

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 9654	. . 3
2		zcn 9654	. . 3
3		apne 8954	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2421	. . 3
6		ztri3or 9692	. . . . . 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 9653	. . . . 5
13		zre 9653	. . . . 5
14		reaplt 8919	. . . . . 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 8178

cr 8179

clt 8361 # cap 8912

cz 9649

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 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297

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 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-inn 9308 df-n0 9569 df-z 9650

This theorem is used by: zltlen 9729 msqznn 9751 qapne 10049 qreccl 10052 seqf1oglem1 10971 nn0opthd 11176 fihashneq0 11249 nnabscl 11883 eftcl 12440 dvdsval2 12576 dvdscmulr 12606 dvdsmulcr 12607 fsumdvds 12628 divconjdvds 12635 gcdn0gt0 12774 lcmcllem 12864 lcmid 12877 3lcm2e6woprm 12883 6lcm4e12 12884 mulgcddvds 12891 divgcdcoprmex 12899 cncongr1 12900 cncongr2 12901 isprm3 12915 pcpremul 13095 pceu 13097 pcmul 13103 pcdiv 13104 pcqmul 13105 dvdsprmpweqle 13139 qexpz 13154 4sqlem11 13203 relogbval 16148 relogbzcl 16149 nnlogbexp 16156 logbgcd1irraplemexp 16165 lgslem1 16285 lgsdilem2 16321 lgsdi 16322 lgsne0 16323 lgseisen 16359