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Theorem difprsn2 3848
 Description: Removal of a singleton from an unordered pair. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
Assertion
Ref Expression
difprsn2 (AB → ({A, B} {B}) = {A})

Proof of Theorem difprsn2
StepHypRef Expression
1 prcom 3798 . . 3 {A, B} = {B, A}
21difeq1i 3381 . 2 ({A, B} {B}) = ({B, A} {B})
3 necom 2597 . . 3 (ABBA)
4 difprsn1 3847 . . 3 (BA → ({B, A} {B}) = {A})
53, 4sylbi 187 . 2 (AB → ({B, A} {B}) = {A})
62, 5syl5eq 2397 1 (AB → ({A, B} {B}) = {A})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ≠ wne 2516   ∖ cdif 3206  {csn 3737  {cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742 This theorem is referenced by: (None)
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