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Theorem difprsn1 3477
 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 2267 . 2 (BAAB)
2 disjsn2 3407 . . . 4 (BA → ({B} ∩ {A}) = ∅)
3 disj3 3249 . . . 4 (({B} ∩ {A}) = ∅ ↔ {B} = ({B} ∖ {A}))
42, 3sylib 127 . . 3 (BA → {B} = ({B} ∖ {A}))
5 df-pr 3357 . . . . . 6 {A, B} = ({A} ∪ {B})
65equncomi 3066 . . . . 5 {A, B} = ({B} ∪ {A})
76difeq1i 3035 . . . 4 ({A, B} ∖ {A}) = (({B} ∪ {A}) ∖ {A})
8 difun2 3279 . . . 4 (({B} ∪ {A}) ∖ {A}) = ({B} ∖ {A})
97, 8eqtri 2042 . . 3 ({A, B} ∖ {A}) = ({B} ∖ {A})
104, 9syl6reqr 2073 . 2 (BA → ({A, B} ∖ {A}) = {B})
111, 10sylbir 125 1 (AB → ({A, B} ∖ {A}) = {B})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1228   ≠ wne 2186   ∖ cdif 2891   ∪ cun 2892   ∩ cin 2893  ∅c0 3201  {csn 3350  {cpr 3351 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357 This theorem is referenced by:  difprsn2  3478
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