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Theorem difprsn1 3494
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 2283 . 2 (BAAB)
2 disjsn2 3424 . . . 4 (BA → ({B} ∩ {A}) = ∅)
3 disj3 3266 . . . 4 (({B} ∩ {A}) = ∅ ↔ {B} = ({B} ∖ {A}))
42, 3sylib 127 . . 3 (BA → {B} = ({B} ∖ {A}))
5 df-pr 3374 . . . . . 6 {A, B} = ({A} ∪ {B})
65equncomi 3083 . . . . 5 {A, B} = ({B} ∪ {A})
76difeq1i 3052 . . . 4 ({A, B} ∖ {A}) = (({B} ∪ {A}) ∖ {A})
8 difun2 3296 . . . 4 (({B} ∪ {A}) ∖ {A}) = ({B} ∖ {A})
97, 8eqtri 2057 . . 3 ({A, B} ∖ {A}) = ({B} ∖ {A})
104, 9syl6reqr 2088 . 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 1242  wne 2201  cdif 2908  cun 2909  cin 2910  c0 3218  {csn 3367  {cpr 3368
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374
This theorem is referenced by:  difprsn2  3495
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