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Theorem difprsn1 3571
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 2339 . 2 (𝐵𝐴𝐴𝐵)
2 disjsn2 3500 . . . 4 (𝐵𝐴 → ({𝐵} ∩ {𝐴}) = ∅)
3 disj3 3332 . . . 4 (({𝐵} ∩ {𝐴}) = ∅ ↔ {𝐵} = ({𝐵} ∖ {𝐴}))
42, 3sylib 120 . . 3 (𝐵𝐴 → {𝐵} = ({𝐵} ∖ {𝐴}))
5 df-pr 3448 . . . . . 6 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
65equncomi 3144 . . . . 5 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
76difeq1i 3112 . . . 4 ({𝐴, 𝐵} ∖ {𝐴}) = (({𝐵} ∪ {𝐴}) ∖ {𝐴})
8 difun2 3358 . . . 4 (({𝐵} ∪ {𝐴}) ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
97, 8eqtri 2108 . . 3 ({𝐴, 𝐵} ∖ {𝐴}) = ({𝐵} ∖ {𝐴})
104, 9syl6reqr 2139 . 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 1289  wne 2255  cdif 2994  cun 2995  cin 2996  c0 3284  {csn 3441  {cpr 3442
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-pr 3448
This theorem is referenced by:  difprsn2  3572
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