Theorem difprsn1 3833

Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)

Assertion

Ref	Expression
difprsn1	|- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Proof of Theorem difprsn1

Step	Hyp	Ref	Expression
1		necom 2496	. 2 |- ( B =/= A <-> A =/= B )
2		df-pr 3696	. . . . . 6 |- { A , B } = ( { A } u. { B } )
3	2	equncomi 3365	. . . . 5 |- { A , B } = ( { B } u. { A } )
4	3	difeq1i 3333	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
5		difun2 3589	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
6	4, 5	eqtri 2253	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
7		disjsn2 3752	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
8		disj3 3561	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
9	7, 8	sylib 122	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
10	6, 9	eqtr4id 2284	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 135	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1398   =/= wne 2412   \ cdif 3208   u. cun 3209   i^i cin 3210   (/)c0 3508   {csn 3689   {cpr 3690

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-pr 3696

This theorem is referenced by:  difprsn2  3834