Theorem diftpsn3 3774

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 3641	. . . 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 3290	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = ( ( { A , B } u. { C } ) \ { C } ) )
4		difundir 3426	. . 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 3640	. . . . . . . . 9 |- { A , B } = ( { A } u. { B } )
7	6	a1i 9	. . . . . . . 8 |- ( ( A =/= C /\ B =/= C ) -> { A , B } = ( { A } u. { B } ) )
8	7	ineq1d 3373	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = ( ( { A } u. { B } ) i^i { C } ) )
9		incom 3365	. . . . . . . . 9 |- ( ( { A } u. { B } ) i^i { C } ) = ( { C } i^i ( { A } u. { B } ) )
10		indi 3420	. . . . . . . . 9 |- ( { C } i^i ( { A } u. { B } ) ) = ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) )
11	9, 10	eqtri 2226	. . . . . . . 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 2460	. . . . . . . . . . 11 |- ( A =/= C <-> C =/= A )
14		disjsn2 3696	. . . . . . . . . . 11 |- ( C =/= A -> ( { C } i^i { A } ) = (/) )
15	13, 14	sylbi 121	. . . . . . . . . 10 |- ( A =/= C -> ( { C } i^i { A } ) = (/) )
16	15	adantr 276	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { C } i^i { A } ) = (/) )
17		necom 2460	. . . . . . . . . . 11 |- ( B =/= C <-> C =/= B )
18		disjsn2 3696	. . . . . . . . . . 11 |- ( C =/= B -> ( { C } i^i { B } ) = (/) )
19	17, 18	sylbi 121	. . . . . . . . . 10 |- ( B =/= C -> ( { C } i^i { B } ) = (/) )
20	19	adantl 277	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { C } i^i { B } ) = (/) )
21	16, 20	uneq12d 3328	. . . . . . . 8 |- ( ( A =/= C /\ B =/= C ) -> ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) ) = ( (/) u. (/) ) )
22		unidm 3316	. . . . . . . 8 |- ( (/) u. (/) ) = (/)
23	21, 22	eqtrdi 2254	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) ) = (/) )
24	8, 12, 23	3eqtrd 2242	. . . . . 6 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) )
25		disj3 3513	. . . . . 6 |- ( ( { A , B } i^i { C } ) = (/) <-> { A , B } = ( { A , B } \ { C } ) )
26	24, 25	sylib 122	. . . . 5 |- ( ( A =/= C /\ B =/= C ) -> { A , B } = ( { A , B } \ { C } ) )
27	26	eqcomd 2211	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } \ { C } ) = { A , B } )
28		difid 3529	. . . . 5 |- ( { C } \ { C } ) = (/)
29	28	a1i 9	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> ( { C } \ { C } ) = (/) )
30	27, 29	uneq12d 3328	. . 3 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = ( { A , B } u. (/) ) )
31		un0 3494	. . 3 |- ( { A , B } u. (/) ) = { A , B }
32	30, 31	eqtrdi 2254	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = { A , B } )
33	3, 5, 32	3eqtrd 2242	1 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   =/= wne 2376   \ cdif 3163   u. cun 3164   i^i cin 3165   (/)c0 3460   {csn 3633   {cpr 3634   {ctp 3635

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-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-tp 3641

This theorem is referenced by: (None)