Theorem diftpsn3 3496

Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.)

Assertion

Ref	Expression
diftpsn3	|- ((A =/= C /\ B =/= C) -> ( {A , B , C } \ { C }) = {A , B })

Proof of Theorem diftpsn3

Step	Hyp	Ref	Expression
1		df-tp 3375	. . . 4 |- {A , B , C } = ( {A , B } u. { C })
2	1	a1i 9	. . 3 |- ((A =/= C /\ B =/= C) -> {A , B , C } = ( {A , B } u. { C }))
3	2	difeq1d 3055	. 2 |- ((A =/= C /\ B =/= C) -> ( {A , B , C } \ { C }) = (( {A , B } u. { C }) \ { C }))
4		difundir 3184	. . 3 |- (( {A , B } u. { C }) \ { C }) = (( {A , B } \ { C }) u. ( { C } \ { C }))
5	4	a1i 9	. 2 |- ((A =/= C /\ B =/= C) -> (( {A , B } u. { C }) \ { C }) = (( {A , B } \ { C }) u. ( { C } \ { C })))
6		df-pr 3374	. . . . . . . . 9 |- {A , B } = ( {A } u. {B })
7	6	a1i 9	. . . . . . . 8 |- ((A =/= C /\ B =/= C) -> {A , B } = ( {A } u. {B }))
8	7	ineq1d 3131	. . . . . . 7 |- ((A =/= C /\ B =/= C) -> ( {A , B } i^i { C }) = (( {A } u. {B }) i^i { C }))
9		incom 3123	. . . . . . . . 9 |- (( {A } u. {B }) i^i { C }) = ( { C } i^i ( {A } u. {B }))
10		indi 3178	. . . . . . . . 9 |- ( { C } i^i ( {A } u. {B })) = (( { C } i^i {A }) u. ( { C } i^i {B }))
11	9, 10	eqtri 2057	. . . . . . . 8 |- (( {A } u. {B }) i^i { C }) = (( { C } i^i {A }) u. ( { C } i^i {B }))
12	11	a1i 9	. . . . . . 7 |- ((A =/= C /\ B =/= C) -> (( {A } u. {B }) i^i { C }) = (( { C } i^i {A }) u. ( { C } i^i {B })))
13		necom 2283	. . . . . . . . . . 11 |- (A =/= C <-> C =/= A)
14		disjsn2 3424	. . . . . . . . . . 11 |- ( C =/= A -> ( { C } i^i {A }) = (/))
15	13, 14	sylbi 114	. . . . . . . . . 10 |- (A =/= C -> ( { C } i^i {A }) = (/))
16	15	adantr 261	. . . . . . . . 9 |- ((A =/= C /\ B =/= C) -> ( { C } i^i {A }) = (/))
17		necom 2283	. . . . . . . . . . 11 |- (B =/= C <-> C =/= B)
18		disjsn2 3424	. . . . . . . . . . 11 |- ( C =/= B -> ( { C } i^i {B }) = (/))
19	17, 18	sylbi 114	. . . . . . . . . 10 |- (B =/= C -> ( { C } i^i {B }) = (/))
20	19	adantl 262	. . . . . . . . 9 |- ((A =/= C /\ B =/= C) -> ( { C } i^i {B }) = (/))
21	16, 20	uneq12d 3092	. . . . . . . 8 |- ((A =/= C /\ B =/= C) -> (( { C } i^i {A }) u. ( { C } i^i {B })) = ( (/) u. (/)))
22		unidm 3080	. . . . . . . 8 |- ( (/) u. (/)) = (/)
23	21, 22	syl6eq 2085	. . . . . . 7 |- ((A =/= C /\ B =/= C) -> (( { C } i^i {A }) u. ( { C } i^i {B })) = (/))
24	8, 12, 23	3eqtrd 2073	. . . . . 6 |- ((A =/= C /\ B =/= C) -> ( {A , B } i^i { C }) = (/))
25		disj3 3266	. . . . . 6 |- (( {A , B } i^i { C }) = (/) <-> {A , B } = ( {A , B } \ { C }))
26	24, 25	sylib 127	. . . . 5 |- ((A =/= C /\ B =/= C) -> {A , B } = ( {A , B } \ { C }))
27	26	eqcomd 2042	. . . 4 |- ((A =/= C /\ B =/= C) -> ( {A , B } \ { C }) = {A , B })
28		difid 3286	. . . . 5 |- ( { C } \ { C }) = (/)
29	28	a1i 9	. . . 4 |- ((A =/= C /\ B =/= C) -> ( { C } \ { C }) = (/))
30	27, 29	uneq12d 3092	. . 3 |- ((A =/= C /\ B =/= C) -> (( {A , B } \ { C }) u. ( { C } \ { C })) = ( {A , B } u. (/)))
31		un0 3245	. . 3 |- ( {A , B } u. (/)) = {A , B }
32	30, 31	syl6eq 2085	. 2 |- ((A =/= C /\ B =/= C) -> (( {A , B } \ { C }) u. ( { C } \ { C })) = {A , B })
33	3, 5, 32	3eqtrd 2073	1 |- ((A =/= C /\ B =/= C) -> ( {A , B , C } \ { C }) = {A , B })

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1242   =/= wne 2201   \ cdif 2908   u. cun 2909   i^i cin 2910   (/)c0 3218   {csn 3367   {cpr 3368   {ctp 3369

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-tp 3375

This theorem is referenced by: (None)