Theorem tpprceq3 4335

Description: An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

Assertion

Ref	Expression
tpprceq3	|- ( -. ( C e. _V /\ C =/= B ) -> { A , B , C } = { A , B } )

Proof of Theorem tpprceq3

Step	Hyp	Ref	Expression
1		ianor 509	. 2 |- ( -. ( C e. _V /\ C =/= B ) <-> ( -. C e. _V \/ -. C =/= B ) )
2		prprc2 4301	. . . . 5 |- ( -. C e. _V -> { B , C } = { B } )
3	2	uneq1d 3766	. . . 4 |- ( -. C e. _V -> ( { B , C } u. { A } ) = ( { B } u. { A } ) )
4		tprot 4284	. . . . 5 |- { A , B , C } = { B , C , A }
5		df-tp 4182	. . . . 5 |- { B , C , A } = ( { B , C } u. { A } )
6	4, 5	eqtri 2644	. . . 4 |- { A , B , C } = ( { B , C } u. { A } )
7		prcom 4267	. . . . 5 |- { A , B } = { B , A }
8		df-pr 4180	. . . . 5 |- { B , A } = ( { B } u. { A } )
9	7, 8	eqtri 2644	. . . 4 |- { A , B } = ( { B } u. { A } )
10	3, 6, 9	3eqtr4g 2681	. . 3 |- ( -. C e. _V -> { A , B , C } = { A , B } )
11		nne 2798	. . . 4 |- ( -. C =/= B <-> C = B )
12		tppreq3 4294	. . . . 5 |- ( B = C -> { A , B , C } = { A , B } )
13	12	eqcoms 2630	. . . 4 |- ( C = B -> { A , B , C } = { A , B } )
14	11, 13	sylbi 207	. . 3 |- ( -. C =/= B -> { A , B , C } = { A , B } )
15	10, 14	jaoi 394	. 2 |- ( ( -. C e. _V \/ -. C =/= B ) -> { A , B , C } = { A , B } )
16	1, 15	sylbi 207	1 |- ( -. ( C e. _V /\ C =/= B ) -> { A , B , C } = { A , B } )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   {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-3or 1038  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-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  tppreqb  4336  1to3vfriswmgr  27144