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Theorem difprsn1 3533
 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 2330 . 2 (𝐵𝐴𝐴𝐵)
2 disjsn2 3463 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
3 disj3 3303 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
42, 3sylib 120 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
5 df-pr 3413 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65equncomi 3119 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
76difeq1i 3087 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
8 difun2 3329 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
97, 8eqtri 2102 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
104, 9syl6reqr 2133 . 2 (𝐵𝐴 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
111, 10sylbir 133 1 (𝐴𝐵 → ({𝐴, 𝐵} ∖ {𝐴}) = {𝐵})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1285   ≠ wne 2246   ∖ cdif 2971   ∪ cun 2972   ∩ cin 2973  ∅c0 3258  {csn 3406  {cpr 3407 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 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-sn 3412  df-pr 3413 This theorem is referenced by:  difprsn2  3534
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