Theorem difprsn1 3849

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 2504	. 2 |- ( B =/= A <-> A =/= B )
2		df-pr 3712	. . . . . 6 |- { A , B } = ( { A } u. { B } )
3	2	equncomi 3375	. . . . 5 |- { A , B } = ( { B } u. { A } )
4	3	difeq1i 3343	. . . 4 |- ( { A , B } \ { A } ) = ( ( { B } u. { A } ) \ { A } )
5		difun2 3604	. . . 4 |- ( ( { B } u. { A } ) \ { A } ) = ( { B } \ { A } )
6	4, 5	eqtri 2259	. . 3 |- ( { A , B } \ { A } ) = ( { B } \ { A } )
7		disjsn2 3768	. . . 4 |- ( B =/= A -> ( { B } i^i { A } ) = (/) )
8		disj3 3576	. . . 4 |- ( ( { B } i^i { A } ) = (/) <-> { B } = ( { B } \ { A } ) )
9	7, 8	sylib 122	. . 3 |- ( B =/= A -> { B } = ( { B } \ { A } ) )
10	6, 9	eqtr4id 2290	. 2 |- ( B =/= A -> ( { A , B } \ { A } ) = { B } )
11	1, 10	sylbir 135	1 |- ( A =/= B -> ( { A , B } \ { A } ) = { B } )

Syntax hints:   -> wi 4   = wceq 1402   =/= wne 2420   \ cdif 3217   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3705   {cpr 3706

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-pr 3712

This theorem is referenced by:  difprsn2  3850