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Theorem difprsn2 3560
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 3501 . . 3 {𝐴, 𝐵} = {𝐵, 𝐴}
21difeq1i 3103 . 2 ({𝐴, 𝐵} ∖ {𝐵}) = ({𝐵, 𝐴} ∖ {𝐵})
3 necom 2335 . . 3 (𝐴𝐵𝐵𝐴)
4 difprsn1 3559 . . 3 (𝐵𝐴 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
53, 4sylbi 119 . 2 (𝐴𝐵 → ({𝐵, 𝐴} ∖ {𝐵}) = {𝐴})
62, 5syl5eq 2129 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐵}) = {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wne 2251  cdif 2985  {csn 3431  {cpr 3432
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rab 2364  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-sn 3437  df-pr 3438
This theorem is referenced by: (None)
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