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Theorem difprsn1 3847
 Description: Removal of a singleton from an unordered pair. (Contributed by Thierry Arnoux, 4-Feb-2017.)
Assertion
Ref Expression
difprsn1 (AB → ({A, B} {A}) = {B})

Proof of Theorem difprsn1
StepHypRef Expression
1 necom 2597 . 2 (BAAB)
2 disjsn2 3787 . . . 4 (BA → ({B} ∩ {A}) = )
3 disj3 3595 . . . 4 (({B} ∩ {A}) = ↔ {B} = ({B} {A}))
42, 3sylib 188 . . 3 (BA → {B} = ({B} {A}))
5 df-pr 3742 . . . . . 6 {A, B} = ({A} ∪ {B})
65equncomi 3410 . . . . 5 {A, B} = ({B} ∪ {A})
76difeq1i 3381 . . . 4 ({A, B} {A}) = (({B} ∪ {A}) {A})
8 difun2 3629 . . . 4 (({B} ∪ {A}) {A}) = ({B} {A})
97, 8eqtri 2373 . . 3 ({A, B} {A}) = ({B} {A})
104, 9syl6reqr 2404 . 2 (BA → ({A, B} {A}) = {B})
111, 10sylbir 204 1 (AB → ({A, B} {A}) = {B})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ≠ wne 2516   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208  ∅c0 3550  {csn 3737  {cpr 3738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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:  difprsn2  3848
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