Theorem diftpsn3 4332

Description: Removal of a singleton from an unordered triple. (Contributed by Alexander van der Vekens, 5-Oct-2017.) (Proof shortened by JJ, 23-Jul-2021.)

Assertion

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

Proof of Theorem diftpsn3

Step	Hyp	Ref	Expression
1		disjprsn 4250	. . . . 5 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } i^i { C } ) = (/) )
2		disj3 4021	. . . . 5 |- ( ( { A , B } i^i { C } ) = (/) <-> { A , B } = ( { A , B } \ { C } ) )
3	1, 2	sylib 208	. . . 4 |- ( ( A =/= C /\ B =/= C ) -> { A , B } = ( { A , B } \ { C } ) )
4	3	eqcomd 2628	. . 3 |- ( ( A =/= C /\ B =/= C ) -> ( { A , B } \ { C } ) = { A , B } )
5		difid 3948	. . . 4 |- ( { C } \ { C } ) = (/)
6	5	a1i 11	. . 3 |- ( ( A =/= C /\ B =/= C ) -> ( { C } \ { C } ) = (/) )
7	4, 6	uneq12d 3768	. 2 |- ( ( A =/= C /\ B =/= C ) -> ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = ( { A , B } u. (/) ) )
8		df-tp 4182	. . . 4 |- { A , B , C } = ( { A , B } u. { C } )
9	8	difeq1i 3724	. . 3 |- ( { A , B , C } \ { C } ) = ( ( { A , B } u. { C } ) \ { C } )
10		difundir 3880	. . 3 |- ( ( { A , B } u. { C } ) \ { C } ) = ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) )
11	9, 10	eqtr2i 2645	. 2 |- ( ( { A , B } \ { C } ) u. ( { C } \ { C } ) ) = ( { A , B , C } \ { C } )
12		un0 3967	. 2 |- ( { A , B } u. (/) ) = { A , B }
13	7, 11, 12	3eqtr3g 2679	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:  f13dfv  6530  nb3grprlem2  26283  cplgr3v  26331  frgr3v  27139  3vfriswmgr  27142  signswch  30638  signstfvcl  30650