zapne - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > zapne		Unicode version

Theorem zapne 9449

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 9379	. . 3
2		zcn 9379	. . 3
3		apne 8698	. . 3 $/\$ #
4	1, 2, 3	syl2an 289	. 2 $/\$ #
5		df-ne 2377	. . 3
6		ztri3or 9417	. . . . . 6 $/\$ $\/$ $\/$
7		3orrot 987	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
8		3orass 984	. . . . . . 7 $\/$ $\/$ $\/$ $\/$
9	7, 8	bitri 184	. . . . . 6 $\/$ $\/$ $\/$ $\/$
10	6, 9	sylib 122	. . . . 5 $/\$ $\/$ $\/$
11	10	ord 726	. . . 4 $/\$ $\/$
12		zre 9378	. . . . 5
13		zre 9378	. . . . 5
14		reaplt 8663	. . . . . 6 $/\$ # $\/$
15		orcom 730	. . . . . 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 710 $\/$ w3o 980

wceq 1373

wcel 2176

wne 2376 class class class wbr 4045

cc 7925

cr 7926

clt 8109 # cap 8656

cz 9374

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4163 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-br 4046 df-opab 4107 df-id 4341 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-iota 5233 df-fun 5274 df-fv 5280 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-inn 9039 df-n0 9298 df-z 9375

This theorem is referenced by: zltlen 9453 msqznn 9475 qapne 9762 qreccl 9765 seqf1oglem1 10666 nn0opthd 10869 fihashneq0 10941 nnabscl 11444 eftcl 11998 dvdsval2 12134 dvdscmulr 12164 dvdsmulcr 12165 fsumdvds 12186 divconjdvds 12193 gcdn0gt0 12332 lcmcllem 12422 lcmid 12435 3lcm2e6woprm 12441 6lcm4e12 12442 mulgcddvds 12449 divgcdcoprmex 12457 cncongr1 12458 cncongr2 12459 isprm3 12473 pcpremul 12649 pceu 12651 pcmul 12657 pcdiv 12658 pcqmul 12659 dvdsprmpweqle 12693 qexpz 12708 4sqlem11 12757 relogbval 15456 relogbzcl 15457 nnlogbexp 15464 logbgcd1irraplemexp 15473 lgslem1 15510 lgsdilem2 15546 lgsdi 15547 lgsne0 15548 lgseisen 15584