Theorem diftpsn3 3661

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 3535	. . . 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 3193	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = ( ( { A , B } u. { C } ) \ { C } ) )
4		difundir 3329	. . 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 3534	. . . . . . . . 9 |- { A , B } = ( { A } u. { B } )
7	6	a1i 9	. . . . . . . 8 |- ( ( A =/= C /\ B =/= C ) -> { A , B } = ( { A } u. { B } ) )
8	7	ineq1d 3276	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = ( ( { A } u. { B } ) i^i { C } ) )
9		incom 3268	. . . . . . . . 9 |- ( ( { A } u. { B } ) i^i { C } ) = ( { C } i^i ( { A } u. { B } ) )
10		indi 3323	. . . . . . . . 9 |- ( { C } i^i ( { A } u. { B } ) ) = ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) )
11	9, 10	eqtri 2160	. . . . . . . 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 2392	. . . . . . . . . . 11 |- ( A =/= C <-> C =/= A )
14		disjsn2 3586	. . . . . . . . . . 11 |- ( C =/= A -> ( { C } i^i { A } ) = (/) )
15	13, 14	sylbi 120	. . . . . . . . . 10 |- ( A =/= C -> ( { C } i^i { A } ) = (/) )
16	15	adantr 274	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { C } i^i { A } ) = (/) )
17		necom 2392	. . . . . . . . . . 11 |- ( B =/= C <-> C =/= B )
18		disjsn2 3586	. . . . . . . . . . 11 |- ( C =/= B -> ( { C } i^i { B } ) = (/) )
19	17, 18	sylbi 120	. . . . . . . . . 10 |- ( B =/= C -> ( { C } i^i { B } ) = (/) )
20	19	adantl 275	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { C } i^i { B } ) = (/) )
21	16, 20	uneq12d 3231	. . . . . . . 8 |- ( ( A =/= C /\ B =/= C ) -> ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) ) = ( (/) u. (/) ) )
22		unidm 3219	. . . . . . . 8 |- ( (/) u. (/) ) = (/)
23	21, 22	syl6eq 2188	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) ) = (/) )
24	8, 12, 23	3eqtrd 2176	. . . . . 6 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) )
25		disj3 3415	. . . . . 6 |- ( ( { A , B } i^i { C } ) = (/) <-> { A , B } = ( { A , B } \ { C } ) )
26	24, 25	sylib 121	. . . . 5 |- ( ( A =/= C /\ B =/= C ) -> { A , B } = ( { A , B } \ { C } ) )
27	26	eqcomd 2145	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } \ { C } ) = { A , B } )
28		difid 3431	. . . . 5 |- ( { C } \ { C } ) = (/)
29	28	a1i 9	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> ( { C } \ { C } ) = (/) )
30	27, 29	uneq12d 3231	. . 3 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = ( { A , B } u. (/) ) )
31		un0 3396	. . 3 |- ( { A , B } u. (/) ) = { A , B }
32	30, 31	syl6eq 2188	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = { A , B } )
33	3, 5, 32	3eqtrd 2176	1 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   =/= wne 2308   \ cdif 3068   u. cun 3069   i^i cin 3070   (/)c0 3363   {csn 3527   {cpr 3528   {ctp 3529

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-tp 3535

This theorem is referenced by: (None)