Theorem diftpsn3 3814

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 3677	. . . 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 3324	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = ( ( { A , B } u. { C } ) \ { C } ) )
4		difundir 3460	. . 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 3676	. . . . . . . . 9 |- { A , B } = ( { A } u. { B } )
7	6	a1i 9	. . . . . . . 8 |- ( ( A =/= C /\ B =/= C ) -> { A , B } = ( { A } u. { B } ) )
8	7	ineq1d 3407	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = ( ( { A } u. { B } ) i^i { C } ) )
9		incom 3399	. . . . . . . . 9 |- ( ( { A } u. { B } ) i^i { C } ) = ( { C } i^i ( { A } u. { B } ) )
10		indi 3454	. . . . . . . . 9 |- ( { C } i^i ( { A } u. { B } ) ) = ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) )
11	9, 10	eqtri 2252	. . . . . . . 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 2486	. . . . . . . . . . 11 |- ( A =/= C <-> C =/= A )
14		disjsn2 3732	. . . . . . . . . . 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 2486	. . . . . . . . . . 11 |- ( B =/= C <-> C =/= B )
18		disjsn2 3732	. . . . . . . . . . 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 3362	. . . . . . . 8 |- ( ( A =/= C /\ B =/= C ) -> ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) ) = ( (/) u. (/) ) )
22		unidm 3350	. . . . . . . 8 |- ( (/) u. (/) ) = (/)
23	21, 22	eqtrdi 2280	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) ) = (/) )
24	8, 12, 23	3eqtrd 2268	. . . . . 6 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) )
25		disj3 3547	. . . . . 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 2237	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } \ { C } ) = { A , B } )
28		difid 3563	. . . . 5 |- ( { C } \ { C } ) = (/)
29	28	a1i 9	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> ( { C } \ { C } ) = (/) )
30	27, 29	uneq12d 3362	. . 3 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = ( { A , B } u. (/) ) )
31		un0 3528	. . 3 |- ( { A , B } u. (/) ) = { A , B }
32	30, 31	eqtrdi 2280	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = { A , B } )
33	3, 5, 32	3eqtrd 2268	1 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   =/= wne 2402   \ cdif 3197   u. cun 3198   i^i cin 3199   (/)c0 3494   {csn 3669   {cpr 3670   {ctp 3671

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-pr 3676  df-tp 3677

This theorem is referenced by: (None)