apneg - Intuitionistic Logic Explorer

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Theorem apneg 8947

Description: Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)

**Assertion**
Ref	Expression
apneg	$/\$ # #

Proof of Theorem apneg Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 8328	. . 3
2	1	adantl 277	. 2 $/\$
3		cnre 8328	. . . . . 6
4	3	ad3antrrr 496	. . . . 5 $/\$ $/\$ $/\$ $/\$
5		simpr 110	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		simpllr 540	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7	5, 6	breq12d 4144	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
8		simplrl 541	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9		simplrr 542	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simprl 535	. . . . . . . . . 10 $/\$ $/\$ $/\$
11	10	ad3antrrr 496	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simprr 537	. . . . . . . . . 10 $/\$ $/\$ $/\$
13	12	ad3antrrr 496	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		apreim 8939	. . . . . . . . 9 $/\$ $/\$ $/\$ # # $\/$ #
15	8, 9, 11, 13, 14	syl22anc 1279	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
16	8	renegcld 8714	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17	9	renegcld 8714	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	11	renegcld 8714	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	13	renegcld 8714	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		apreim 8939	. . . . . . . . . 10 $/\$ $/\$ $/\$ # # $\/$ #
21	16, 17, 18, 19, 20	syl22anc 1279	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # # $\/$ #
22	8	recnd 8360	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23		ax-icn 8280	. . . . . . . . . . . . . 14
24	23	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	9	recnd 8360	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	24, 25	mulcld 8352	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	22, 26	negdid 8657	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	5	negeqd 8528	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	24, 25	mulneg2d 8746	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29	oveq2d 6104	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	27, 28, 30	3eqtr4d 2281	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	11	recnd 8360	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	13	recnd 8360	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	24, 33	mulcld 8352	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	32, 34	negdid 8657	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	6	negeqd 8528	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	24, 33	mulneg2d 8746	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	37	oveq2d 6104	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	35, 36, 38	3eqtr4d 2281	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	31, 39	breq12d 4144	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
41		reapneg 8933	. . . . . . . . . . 11 $/\$ # #
42	8, 11, 41	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
43		reapneg 8933	. . . . . . . . . . 11 $/\$ # #
44	9, 13, 43	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
45	42, 44	orbi12d 805	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $\/$ # # $\/$ #
46	21, 40, 45	3bitr4rd 221	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # $\/$ # #
47	7, 15, 46	3bitrd 214	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
48	47	ex 115	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ # #
49	48	rexlimdvva 2676	. . . . 5 $/\$ $/\$ $/\$ $/\$ # #
50	4, 49	mpd 13	. . . 4 $/\$ $/\$ $/\$ $/\$ # #
51	50	ex 115	. . 3 $/\$ $/\$ $/\$ # #
52	51	rexlimdvva 2676	. 2 $/\$ # #
53	2, 52	mpd 13	1 $/\$ # #

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209

wrex 2529 class class class wbr 4131 (class class class)co 6088

cc 8183

cr 8184

ci 8187

caddc 8188

cmul 8190

cneg 8505 # cap 8917

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 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302

This proof depends on definitions: df-bi 117 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-br 4132 df-opab 4194 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-iota 5340 df-fun 5382 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-pnf 8368 df-mnf 8369 df-ltxr 8371 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918

This theorem is used by: mulext1 8948 negap0 8966 div2subap 9175 cjap 11694 geosergap 12298

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