zapne - Intuitionistic Logic Explorer

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Theorem zapne 9286

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 9217	. . 3
2		zcn 9217	. . 3
3		apne 8542	. . 3 $/\$ #
4	1, 2, 3	syl2an 287	. 2 $/\$ #
5		df-ne 2341	. . 3
6		ztri3or 9255	. . . . . 6 $/\$ $\/$ $\/$
7		3orrot 979	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		3orass 976	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
9	7, 8	bitri 183	. . . . . 6 $\/$ $\/$ $\/$ $\/$
10	6, 9	sylib 121	. . . . 5 $/\$ $\/$ $\/$
11	10	ord 719	. . . 4 $/\$ $\/$
12		zre 9216	. . . . 5
13		zre 9216	. . . . 5
14		reaplt 8507	. . . . . 6 $/\$ # $\/$
15		orcom 723	. . . . . 6 $\/$ $\/$
16	14, 15	bitrdi 195	. . . . 5 $/\$ # $\/$
17	12, 13, 16	syl2an 287	. . . 4 $/\$ # $\/$
18	11, 17	sylibrd 168	. . 3 $/\$ #
19	5, 18	syl5bi 151	. 2 $/\$ #
20	4, 19	impbid 128	1 $/\$ #

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 $\/$ w3o 972

wceq 1348

wcel 2141

wne 2340 class class class wbr 3989

cc 7772

cr 7773

clt 7954 # cap 8500

cz 9212

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891

This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-inn 8879 df-n0 9136 df-z 9213

This theorem is referenced by: zltlen 9290 msqznn 9312 qapne 9598 qreccl 9601 nn0opthd 10656 fihashneq0 10729 nnabscl 11064 eftcl 11617 dvdsval2 11752 dvdscmulr 11782 dvdsmulcr 11783 divconjdvds 11809 gcdn0gt0 11933 lcmcllem 12021 lcmid 12034 3lcm2e6woprm 12040 6lcm4e12 12041 mulgcddvds 12048 divgcdcoprmex 12056 cncongr1 12057 cncongr2 12058 isprm3 12072 pcpremul 12247 pceu 12249 pcmul 12255 pcdiv 12256 pcqmul 12257 dvdsprmpweqle 12290 qexpz 12304 relogbval 13663 relogbzcl 13664 nnlogbexp 13671 logbgcd1irraplemexp 13680 lgslem1 13695 lgsdilem2 13731 lgsdi 13732 lgsne0 13733