Theorem diftpsn3OLD 4333

Description: Obsolete proof of diftpsn3 4332 as of 23-Jul-2021. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
diftpsn3OLD	|- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )

Proof of Theorem diftpsn3OLD

Step	Hyp	Ref	Expression
1		df-tp 4182	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
2	1	a1i 11	. . 3 |- ( ( A =/= C /\ B =/= C ) -> { A , B , C } = ( { A , B } u. { C } ) )
3	2	difeq1d 3727	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = ( ( { A , B } u. { C } ) \ { C } ) )
4		difundir 3880	. . 3 |- ( ( { A , B } u. { C } ) \ { C } ) = ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) )
5	4	a1i 11	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } u. { C } ) \ { C } ) = ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) )
6		df-pr 4180	. . . . . . . . 9 |- { A , B } = ( { A } u. { B } )
7	6	a1i 11	. . . . . . . 8 |- ( ( A =/= C /\ B =/= C ) -> { A , B } = ( { A } u. { B } ) )
8	7	ineq1d 3813	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = ( ( { A } u. { B } ) i^i { C } ) )
9		incom 3805	. . . . . . . . 9 |- ( ( { A } u. { B } ) i^i { C } ) = ( { C } i^i ( { A } u. { B } ) )
10		indi 3873	. . . . . . . . 9 |- ( { C } i^i ( { A } u. { B } ) ) = ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) )
11	9, 10	eqtri 2644	. . . . . . . 8 |- ( ( { A } u. { B } ) i^i { C } ) = ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) )
12	11	a1i 11	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A } u. { B } ) i^i { C } ) = ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) ) )
13		necom 2847	. . . . . . . . . . 11 |- ( A =/= C <-> C =/= A )
14		disjsn2 4247	. . . . . . . . . . 11 |- ( C =/= A -> ( { C } i^i { A } ) = (/) )
15	13, 14	sylbi 207	. . . . . . . . . 10 |- ( A =/= C -> ( { C } i^i { A } ) = (/) )
16	15	adantr 481	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { C } i^i { A } ) = (/) )
17		necom 2847	. . . . . . . . . . 11 |- ( B =/= C <-> C =/= B )
18		disjsn2 4247	. . . . . . . . . . 11 |- ( C =/= B -> ( { C } i^i { B } ) = (/) )
19	17, 18	sylbi 207	. . . . . . . . . 10 |- ( B =/= C -> ( { C } i^i { B } ) = (/) )
20	19	adantl 482	. . . . . . . . 9 |- ( ( A =/= C /\ B =/= C ) -> ( { C } i^i { B } ) = (/) )
21	16, 20	uneq12d 3768	. . . . . . . 8 |- ( ( A =/= C /\ B =/= C ) -> ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) ) = ( (/) u. (/) ) )
22		unidm 3756	. . . . . . . 8 |- ( (/) u. (/) ) = (/)
23	21, 22	syl6eq 2672	. . . . . . 7 |- ( ( A =/= C /\ B =/= C ) -> ( ( { C } i^i { A } ) u. ( { C } i^i { B } ) ) = (/) )
24	8, 12, 23	3eqtrd 2660	. . . . . 6 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) )
25		disj3 4021	. . . . . 6 |- ( ( { A , B } i^i { C } ) = (/) <-> { A , B } = ( { A , B } \ { C } ) )
26	24, 25	sylib 208	. . . . 5 |- ( ( A =/= C /\ B =/= C ) -> { A , B } = ( { A , B } \ { C } ) )
27	26	eqcomd 2628	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } \ { C } ) = { A , B } )
28		difid 3948	. . . . 5 |- ( { C } \ { C } ) = (/)
29	28	a1i 11	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> ( { C } \ { C } ) = (/) )
30	27, 29	uneq12d 3768	. . 3 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = ( { A , B } u. (/) ) )
31		un0 3967	. . 3 |- ( { A , B } u. (/) ) = { A , B }
32	30, 31	syl6eq 2672	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = { A , B } )
33	3, 5, 32	3eqtrd 2660	1 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B , C } \ { C } ) = { A , B } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179   {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by: (None)