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Theorem difprsn2 3545
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 3486 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21difeq1i 3096 . 2 ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵})
3 necom 2333 . . 3 (𝐴𝐵𝐵𝐴)
4 difprsn1 3544 . . 3 (𝐵𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
53, 4sylbi 119 . 2 (𝐴𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
62, 5syl5eq 2127 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wne 2249  cdif 2979  {csn 3416  {cpr 3417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-sn 3422  df-pr 3423
This theorem is referenced by: (None)
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